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Grabación_Cipas 2_Fase 4_Probabilidad y estadística_Libia Acosta — Transcript

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0:08Good afternoon, dear students.
0:11Today we have a small audience.
0:14Let's see if the students are joining in
0:17and we're going to start with an
0:19explanation of distributions.
0:23Okay? Can you hear me well?
0:29Yes, teacher, I can hear you.
0:31>> Okay, thank you, Andi.
0:33So, look, here we are starting, if you
0:36see the image, with the invitation to
0:39the tenth virtual statistics seminar,
0:42which will be next week.
0:46So, remember to stay tuned because there
0:48is an activity that must be developed
0:51with a conference that will be given in
0:53this seminar.
0:57Well, for today we are going to study
0:59continuous distributions.
1:02Well, there are many continuous
1:04distributions, but we are going to focus
1:07on two of them.
1:09Remember that when we talk about a
1:12continuous variable, it is when we
1:14observe that any value can occur within
1:17an interval.
1:19Yes? For example, if we say salary, then
1:23a person's salary could be 2,500 x 40.
1:27Yes?
1:28So, in that way we would say that it is
1:31continuous. Yes?
1:33or for example, when we weigh a product
1:35, then the weight of the product is a
1:37continuous random variable.
1:39So here we are going to see some of the
1:42normal distribution.
1:43So, notice that here we have a bell
1:45curve from Gus.
1:48Every time we graph the data, it can be
1:52through a histogram, we smooth it and
1:55observe that we find a bell curve.
2:00It is telling me that my data is
2:01normally distributed. Yes, for example,
2:04even salaries could be normally
2:06distributed.
2:08We would have to make the graph to check
2:10it.
2:12So, this is a continuous distribution
2:15that is going to be bell-shaped and
2:18symmetrical with respect to the mean.
2:22Distribution is this.
2:24There is the mean.
2:25And that mean that we have here,
2:33this mean, also the median and the mode
2:36are the same. Okay? Good.
2:40Asymptotic means that it sticks very
2:42close to the X-axis, but it will never
2:44touch it.
2:46Yes, the same for either of the two
2:48tails.
2:50The area under the curve, that is, and
2:53on the X-axis, all of this is going to
2:55be equal to 1 or 100%of the data, which
2:58would be written as the probability that
3:01the x's are between negative infinity
3:04and infinity, it's going to be one.
3:07Since the mean that's here in the middle
3:11, here is my mu.
3:13Everything that's below the mean is 50%
3:17of the data and the other 50 will be
3:20above the mean.
3:23Right?
3:24In other words, if I were to fold that
3:27little bell, I'd have two equal parts
3:29and each one would cover 50%of the data
3:32.
3:34We're always going to depend on two
3:37parameters, the mean and the standard
3:39deviation. Yes, they're important
3:42parameters.
3:43Here X represents the variable that
3:46we're studying.
3:48It could be salary, age, the weight of a
3:51product.
3:52Well, anyway.
3:54This loop means normally distributed
3:57with mean mu and standard deviation
4:00sigma.
4:01Let's see some distributions here.
4:06Look, here we have a normal distribution
4:10in red, another in green, and another in
4:14blue.
4:16All three are normal distributions.
4:17Notice that they are bell-shaped and
4:20symmetrical with respect to their mean,
4:23but they have one thing in common.
4:27What could the three distributions have
4:28in common?
4:29Who knows?
4:38That the variability, well, the height
4:42is the same. Very good.
4:46Notice that the variability, that is,
4:48the standard deviation, this value that
4:51we show here is the same, because if we
4:54place one on top of the other, the
4:56dispersion is the same. In other words,
4:59the standard deviation is the same for
5:02the three, but notice that the means are
5:05different. Here it is zero, that is, the
5:08smallest mean is 1, that of the mean of
5:11the red distribution.
5:14And the largest is that of distribution
5:18three.
5:19Why the blue one?
5:20Because it is further from zero, right?
5:24And here we have the cutoff at zero, so
5:26everything that is further to the right
5:29is larger and everything that is further
5:32to the left, close to zero, is smaller.
5:35Yes? So, notice, the three distributions
5:37are normal.
5:39Uh, the three distributions have the
5:42same standard deviation, the same
5:45variability, but they differ in terms of
5:48the mean.
5:51Okay, and here, what can we see in these
5:53three distributions?
5:59Who can tell me what we see the same and
6:02what we see differently?
6:07Note that all three are bell-shaped, but
6:09they are not the same bell.
6:11There are some that have greater
6:13dispersion, greater volatility of the
6:15data.
6:16Look, the red one is more flattened, but
6:19it has quite a lot of dispersion.
6:24And the one with the least dispersion or
6:26standard deviation is the green one,
6:28because the data is closer to the mean.
6:31Yes.
6:33And the mean is the same for all three
6:35because they have the same center. Do
6:38you see? So, the difference between the
6:41previous one, where in this one we had
6:43the same deviation.
6:46And here we had different deviations,
6:49but the same mean.
6:51Well, we are always going to find in a
6:54normal distribution. Look, here is the
6:57mean, the center of the distribution,
6:59but the normal distribution is always
7:01divided into deviations.
7:04One deviation to the left, two
7:06deviations to the left, and three
7:08deviations to the left.
7:11So, it means that I subtract one from
7:13the mean, I subtract two deviations from
7:16the mean, and finally I subtract two
7:18deviations from the mean.
7:20And there is still a small probability,
7:23remember the property of being
7:25asymptotic.
7:27And I go with the mean.
7:30I take the mean, I increase it by one
7:33deviation or I add one deviation.
7:36I add two deviations to the mean and I
7:38add three deviations to the mean.
7:41Well, what happens?
7:45If I take this interval, the mean minus
7:48one deviation and the mean plus one
7:51deviation, this interval has 68%of the
7:55data, a lot of data.
7:58But if I move up to two deviations to
8:01the left and two to the right, almost
8:03all the data is found, 95%.
8:08And if I go up to three deviations to
8:11the left, three to the right, and shade
8:14all that, there I find almost all the
8:17data, 99.7%
8:21of the data.
8:23That is, a very small probability in the
8:25tails, that will always be true.
8:28Well, what interests us more here? Hm.
8:32Let's see, this, this here.
8:37So, we are going to calculate
8:38probabilities to the left of a value
8:44and how it is symbolized. Here we have
8:47the value two.
8:49So, wait a second because I have some
8:51...Well, yes, we can review those.
8:55Probability, but the one we are going to
8:57see is not standard.
8:59Wait a second.
9:01This is it. Wait, wait a second.
9:05No, here it is.
9:09Well, let's look at those.
9:10Let's look at those that are useful
9:12anyway.
9:13I mean, what is on the left will be
9:16given to me directly by the function.
9:19Now we are going to see the function.
9:20Yes, directly.
9:22I mean, I type the function, I say the
9:24limit point, which in this case is two,
9:27which is a, and I say accumulated, that
9:29is, it sweeps out everything that is on
9:32the left.
9:33Yes, but what would happen if the value
9:36is on the right and they ask me for the
9:38area on the right, for example, of 0.65
9:41?
9:43Well, they are telling me what the
9:45probability is that the z's in this case
9:48are greater than 0.65.
9:51So, remember that the area under the
9:54curve is a total of one and I would have
9:57to subtract what is on the left, right?
10:00In other words, we would have to
10:03calculate what is on the left, right?
10:06And then I subtract what is on the left
10:09from the total area to give me what is
10:11on the right.
10:13Well, if it is between two points,
10:15notice that here it is between two
10:18points, so what would I do?
10:21So, first I calculate the largest area.
10:24So, here it is 1.5.
10:27So, through the function, if I tell it
10:29the function of the value from 1.5 to
10:32the left, then I have to give it the
10:34cumulative value.
10:36So, it would be like the probability
10:38that Z is less than 1.5.
10:42I give it the function and I tell it 1.5
10:44accumulated and it even gives me this
10:46little white value.
10:49So, we would have to remove the little
10:51white value, the little white area.
10:53So I say, well, it's to calculate the
10:56probability that the zs in this case are
10:58less than -2.
11:01So, how do we calculate this blank area
11:03?
11:04So, it would be, uh, the function, the
11:08point that is -2 accumulated. So it's
11:11going to sweep out all the white.
11:14And now I have these two areas.
11:17So I say to the blue area that includes
11:19the blue and the white, I remove the
11:22white.
11:23So 0.93-0.023.
11:28Yes.
11:29Well, that's the normal distribution.
11:32Now, let's see the example.
11:34And here we have the uniform
11:37distribution.
11:41They are just characteristics so that if
11:44you chose a, for example, the normal one
11:46, you already have characteristics to
11:49build your distribution.
11:52And now the continuous uniform
11:54distribution.
11:57It says, "It is useful for describing
11:59variables with constant probability."
12:02So, that's like a rectangle where we
12:05have the base, but the height is the
12:08probability of occurrence of that
12:11rectangle.
12:13The probability of an event only depends
12:17on the width. So how wide is it?
12:20Let's say that if I have a rectangle,
12:22I'm going to take a small piece of the
12:24rectangle and say up to here, no further
12:26.
12:27So it's going to go from the minimum to
12:29the value. Yes, we already saw how it
12:32would be calculated.
12:34Oh, I lost it.
12:38I lost it. Here it is.
12:40So, it's always going to depend on the
12:43width of the interval considered. Let's
12:46see.
12:47What are the characteristics of this
12:50distribution? It's going to depend on
12:52two parameters, A and B, which determine
12:55where the rectangle goes from to where.
12:59Ready? Good.
13:02The value that the function takes on the
13:05interval AB that we just defined is
13:07constant.
13:09In other words, the probability is going
13:12to be 1 over b-a.
13:14I'll explain, don't worry.
13:17So, the function is going to be 1 over
13:19b-a in the closed interval, in other
13:21words, that includes a and includes b.
13:25Nothing else can happen. What does that
13:27mean?
13:28Look, here is the same thing we just
13:31said and then the X axis and the f dex
13:34determines the probability.
13:37So I say I'm going to go from A to B.
13:41Right?
13:43So if I don't want to start from zero, I
13:45have to take B-A.
13:47Yes, to know what this width is. Yes,
13:51this length, sorry, this length.
13:55So, and the probability is going to be
13:57one, which is this, over that width,
14:00right?
14:011 over b-a.
14:03That's the probability. Right?
14:06So, we're always going to work with a
14:09rectangle and that rectangle is going to
14:11tell me up to where.
14:14For example, I can say up to here, no
14:17further.
14:19Let's say it's a point C. Right?
14:22So that means that the area that I'm
14:24going to sweep is going to be this, no
14:25further, not all of it. Yes.
14:27So I say, ah, well, I have to calculate
14:29this little bit. So I say C, this little
14:35bit m C-A.
14:38Let's write it again.
14:43Let's see, this little piece up to here
14:50measures CA because from here from zero
14:56to C is
15:00from 0 to A is A and notice that I only
15:02need the shaded part. So it will be all
15:06CA. Yes.
15:07And then I would say this is the
15:11rectangle that has base CA and height
15:15B-A, right? So it would be CA over B-A.
15:19Yes? That would be the rectangle or the
15:22probability of this rectangle.
15:25So I say the probability that X is less
15:29than CA. So I say, that's what we just
15:33calculated.
15:36CA-C and from that I say all that is
15:40equal to B-A.
15:42Yes.
15:43So, this is the area to calculate, but
15:46we start from the entire rectangle. Yes
15:49?
15:50So, that's why we always have to put
15:52this denominator B-A, right?
15:55Which is what we should calculate to
15:57find the exercises. Let's go, let's read
16:00an exercise to see what the statement
16:03could be like.
16:07It says: "The time measured in minutes
16:09that a company takes to deliver a food
16:11order.
16:13Remember that time is a continuous
16:16variable and one would say, well, it
16:18could be a normal or a uniform
16:21distribution; but notice that here they
16:24already tell me that it is uniformly
16:26distributed in the interval 50 to 90,
16:29that is, that the rectangle, the base,
16:32goes from 50 to 90." Let's see, I'm
16:35going to try to make a graph here so
16:37that we can see it, it's not going to be
16:40very pretty.
16:42So, remember that here is zero.
16:45Let's say that 50 is over here and 90 is
16:49over there. Okay? So, this is supposed
16:52to have a height, the rectangle that I'm
16:55going to build.
16:59And that rectangle, let's say that each
17:02little piece is going to have a
17:04probability.
17:06So I say that it's going to be 1 over
17:10b-a, that is, 90-50 which gives me 40.
17:13Okay?
17:15So if I'm going to calculate a little
17:17piece, for example, I say, I only want
17:20this little piece.
17:21How much is this worth?
17:24Let's say 60. So, if I'm going to
17:26calculate the probability that x is less
17:29than 60, what do I have to do?
17:33So I say, remember what we explained
17:36before, this whole piece is worth 60,
17:39from 0 to 60 it's worth 60, from 0 to 50
17:43, but I need it to be from 50 to 60. So
17:46I need to measure this little piece.
17:52So it will be 60-50 which gives me 10,
17:55right?
17:58So it would be 10 over 1 over 40*1 over
18:0240 which would be 40. I cancel the zero
18:06and that gives me 1/4 and the
18:09probability is 0.25
18:13which represents 25%.
18:17Yes? So I would say that
18:2125%of orders take less than 60 minutes
18:25to be delivered.
18:27If you see, I did a different exercise.
18:30Here they ask others, what is the
18:32probability that the next delivery will
18:35be late?
18:36Maximum 55. So, I'll have to look for 55
18:40and do the same thing I was doing. Ready
18:44?
18:45That's like a preview of the exercise
18:47we're going to do.
18:49So, let's look at them.
18:52Okay, let's start with approach two.
18:57Remember the normal distribution,
19:00symmetrical bell curve. Yes. Okay, so,
19:03well, here we have our data, but I'm not
19:07going to take all of it. I'm going to
19:11remove a few. Let's see, how many did I
19:13remove?
19:151, 2, 3, 4, 5 So it would be up to 106
19:24because if it's not for
19:27you to do your exercise and it won't
19:31match, don't worry.
19:35Ready. So I have it here.
19:38So, I'm going to delete this, I'm going
19:41to delete this, I'm going to calculate
19:43the mean. Do you remember that the mean
19:46is an average function, right?
19:48So we say average, I select all the data
19:52, that's it.
19:54And enter. And I check F2.
19:58There is the average and it has selected
20:01all the data.
20:05The average, remember that it is the
20:06same mean.
20:07Deviation.
20:09So, I can call it point M because it is
20:12a sample, that is why it is point M. If
20:16it is point P it is because it is a
20:18population
20:22of the data. Remember that our variable
20:25is hours of work experience.
20:28Oh, well, so now I have
20:35the mean, the standard deviation, I am
20:37going to use both of them. Well, what
20:39function am I going to use? The function
20:42that is going to be used is distrip.nor
20:44.
20:46Okay, now we are going to use it. Let's
20:48delete all this.
20:50Oh, God. Delete, delete.
20:53So it says, "What is the probability
20:55that a student has less than 1000 hours
20:58of work experience?
21:01In other words, less goes to the left.
21:03It's like this little graph, that is, I
21:06have to find this little purple part.
21:09So, first I define x, which is 1000.
21:12Then I'm going to calculate the
21:14probability. So, I need the function.
21:18So I write distri pun dot nor and there
21:22all the normals dot nor dot n start to
21:26appear, the standard because we haven't
21:31standardized district.nor.n pun n we
21:34select it we give it the value of x
21:41comma the mean
21:45comma the standard deviation comma and
21:49since we want the cumulative we tell it
21:53true.
21:55Yes?
21:57Good. And that will calculate the area
22:00on the left, this purple. In other words
22:03, if I
22:05put, for example, this here, I say that
22:10it is a probability of 94.4
22:16approximately. So, what can I say?
22:19That there is a 94.39%chance
22:25that what?
22:27that a student has
22:33less than 1000 hours of work experience
22:36.
22:36Yes, less, but if they asked me more,
22:39then it says, here they asked me less.
22:42It is always important to see if it is
22:45less, it would be this answer that is
22:47what they ask us here.
22:49Yes, but if they asked me more, how do I
22:52calculate what is greater?
22:54Remember, remember that the area under
22:58the curve on the X-axis is going to be
23:01100%of the data, which in probability
23:04one.
23:05Yes?
23:06And so I do that. So I say, from 100%
23:11of the data I am going to subtract what
23:14it gave me, what is on the left.
23:17In other words, here on the right I am
23:19going to draw an arrow here to indicate
23:22that this that is in white
23:26represents
23:285 percent. Oh, God, this moved.
23:33Let's see, 5.6
23:37percent of the data.
23:39Okay?
23:40Notice that if I add these two together
23:43, it gives me 100%of the data. Okay?
23:46Good, then yes, we do understand.
23:52Yes, yes, yes.
23:54Or regular, regular.
23:56Let's look at the other one.
23:57>> Yes, sir.
23:58>> Okay. Good, very good, Cristina.
24:01So, if Annie or any of the three of us,
24:04Edison, have any questions, please, I'm
24:07listening.
24:11Well, then let's
24:14remember that if you chose statement A,
24:18then write your name here.
24:21So, for all the exercises, you have to
24:24develop statement A. If Annie, for
24:26example, chose B, then here you must
24:28choose your, write your name and for
24:30each of the following paragraphs, do the
24:33same.
24:35Well, continuing, it says, what is the
24:38probability that a student has less than
24:40500
24:44hours of work experience? So, the
24:46reference point x is 500. Notice that
24:49they tell me that it is less, that is,
24:51that the answer here is here.
24:54probability that x is less than 500. So
24:57, how do we calculate it? So, again we
25:00write distream dot nor y dot n, the
25:05second function that appears.
25:09So, what is x?
25:11500 comma.
25:14the mean, comma, standard deviation and
25:18cumulative because I need cumulative on
25:22the left.
25:24And there you have it, it's ready.
25:27So, now I can interpret because they
25:29asked me the least.
25:31So, the interpretation could be that
25:3732.38%
25:3938%
25:43of the students
25:49have
25:52less than 500 hours of work experience.
26:08Done.
26:10So, if I want to calculate what is
26:13greater, then let's say, remember, it's
26:16100%which is the whole bell and I
26:19subtract what's on the left, which is
26:22approximately 32%.
26:24In other words, there is more area,
26:27there are more students who have more
26:30than 500 hours of work experience, right
26:33?
26:34Well, now it says, what is the
26:37probability that a student has more than
26:40700 hours of work experience?
26:44I take X is 700.
26:48Hm.
26:50Here we can say f (x) is like the
26:54cumulative and that represents the
26:57probability that x is less than 700.
27:01Okay?
27:04We can write it like this or like this.
27:08And that is telling me the same thing.
27:10Capital F is like the cumulative
27:12distribution, okay?
27:15So let's calculate it.
27:17So, be careful, because here they told
27:19me that more.
27:20So, in any case, I have to calculate
27:22everything that is on the left and then
27:25everything that is on the right. Yes,
27:27this is smaller, but what they're asking
27:30me is this.
27:32Probability that x is greater than 700.
27:36That's what they're really asking me.
27:41This one.
27:44Let's put it here in yellow.
27:47So, we calculate what's on the left
27:49first.
27:51So, it's the distribution function point
27:54.
27:57N.
27:59So, what do we tell it?
28:01700
28:03with
28:06the mean, comma the deviation,
28:11cumulative comma.
28:13So
28:15the cumulative
28:21is true.
28:24So, this is what's on the left, but what
28:26I need is what's on the right.
28:29So it would be 100%
28:33F2 minus what's on the left.
28:39It didn't happen.
28:40Let's see, we already corrected it, we
28:42didn't write it as a function.
28:45Plus 100 minus the previous gives me 35.
28:50This is the one we need.
28:53Interpretation. So it would be that the
28:5935.9%
29:05of students
29:14have
29:17more than
29:20700 hours of work experience.
29:27Ready?
29:29So, notice that it's one thing when they
29:32ask me for less, which is directly
29:34through the function, and another when
29:36they ask me for more.
29:40So, I have to remove it, subtract the
29:43total area, which is 100, which is on
29:47the left, and that gives me the area
29:50above, right?
29:52Of 700. Okay, now here is the area
29:55between two values. Let's see, one
29:58second.
30:00Let's see if I can steal the bell here.
30:04Where is it? Where is it? I would close
30:06it.
30:08Here it is.
30:10I'm going to steal this bell to see if I
30:11can steal it. No, I can't because it's
30:14in PDF.
30:17I can't steal it. Okay,
30:27so here we're going to write down this
30:31bell and we're going to remove it. Here
30:35we're going to put a reference point.
30:39Here it tells me between 400 and 800. So
30:42I'm going to put a point here and I say
30:45, here is 400.
30:48And here is 800.
30:52Or So, the area that I'm going to
30:54calculate is this.
30:59Okay, so it says," What is the
31:01probability that a student has between
31:05400 and 800
31:08hours of work experience? "First we take
31:11the values of 400
31:15and 800. Ready?
31:17And now we are going to calculate the
31:19cumulative function.
31:21Remember that here it would be the
31:23probability that x is less than 400.
31:29Yes.
31:37And this would be the largest, right?
31:40We remove the sign, we put a greater.
31:46Well, then the cumulative function is
31:48capitalized F and the cumulative
31:50function is on the left. In other words
31:53, if I take 400, it will calculate
31:56everything that is below 400. Right?
32:00So I say nor.n distribution whose value
32:06is 400 comma
32:11I go to the mean deviation comma the
32:15standard deviation comma the
32:20true value and that's it.
32:24And now, uh, then I am going to
32:26calculate the area that is on the right
32:28. That is for doing a lot of
32:30calculations, but you are not asking me
32:32. For this I would not need this value
32:35or this one.
32:37So, but we are going to calculate it.
32:40So, remember that this calculated
32:43everything that is below 400. This value
32:46would calculate everything that is above
32:49400. Then it would be 100
32:57minus
32:59Ah, no, this is not it, they left it in
33:00probability, let's give it in
33:01probability. So if we keep it in
33:04probability it would be 1 minus this.
33:09Do you see? Then one can express it in
33:12percentage or in probability.
33:15Yes.
33:17So, if everything is worked on in
33:18probability in the exercise, everything
33:21must be worked on in probability. If
33:23everything is worked on in percentage,
33:24everything in percentage. Yes.
33:26Well, let's work on this in probability
33:29.
33:30So, now I'm going to work for 800.
33:35So, in 800 I'm going to calculate the
33:37area below 800, all the purple.
33:42So it would be again a normed
33:45distribution
33:51of which the value 800 comma
33:56the mean, comma the deviation and comma
34:01true. Why?
34:03Because I need the cumulative, right?
34:06And enter.
34:07So, the portion that falls below 800 is
34:10approximately 78 percent, is it not?
34:14Because this is probability.
34:16So, now we are going to calculate what
34:18is above 800, that is, this little black
34:21thing. So, it would be 1 minus the area
34:26on the left so that it gives me the area
34:30on the right. Yes.
34:33So, how does this value come out? Let's
34:35erase it.
34:37So, if I want to calculate between 800
34:40and 400, I say everything less than 800
34:43, that is, this minus everything less
34:46than 400. So, let's do it.
34:50So, everything less than 800, notice
34:52that everything less than 800 sweeps all
34:54this, we would have to subtract
34:56everything less than 400 minus
34:58everything less than 400. And it gives
35:00me the same.
35:02So, the area has already been converted
35:04to a percentage. Notice that it is
35:06already written there as a percentage,
35:09it is already oriented so that it
35:11calculates the percentage.
35:14So, I could say that 59%
35:19of students
35:25have
35:28between 400
35:31and 800
35:34hours
35:37of work experience.
35:43Well, let's do this.
35:46It says," What is the probability that a
35:48student has less than 300 hours of work
35:51experience?
35:53The reference point is 300. So, I'm
35:56going to delete the initial graph, which
35:59is this one.
36:06And I'm going to put 300. So I say,
36:09let's say that 300 is here
36:17and it tells me everything less than
36:19300. So it's going to be everything
36:20that's less than 300. Right?
36:25So, notice that there I would only use
36:27the first one, no more distri.NOR
36:32point n
36:34of 300 comma
36:38the mean, comma the deviation, comma
36:41true because it's the accumulated and
36:44that's it.
36:48I don't need to calculate anything else
36:50because this would be the largest and
36:52here they're asking me to calculate the
36:54least than 300. Yes.
36:57So, notice that here I calculated many
37:00things that I shouldn't have, for
37:02example, I didn't need to calculate this
37:04, this neither, that was like to look at
37:06other things, but here what I really
37:08needed to calculate are these two to
37:10obtain this.
37:13Well, and we would need the
37:15interpretation. So, we would say that
37:1910.11%11%
37:23of the students
37:26have less than 300 hours of work
37:32experience.
37:35Okay, let's go with the last one. So,
37:39the last one tells me the area between
37:42two points.
37:44So, for the area between two points,
37:47look, here when they show me two boxes,
37:51it says, what is the probability that a
37:54student has between 600 and 900 hours of
37:58work experience?
38:00Between 600
38:03and 900 hours of work experience.
38:07So, here we could look for where 300 is
38:10.
38:12Let's say it's around here. And where
38:14does it fall? H,
38:17sorry,
38:22600 over here, let's say,
38:27and 900, let's say it's here.
38:32The area to calculate
38:35would be between those two points.
38:39Okay, so if I calculate 900 with the
38:42function, it sweeps out everything
38:46that's to the left.
38:48If I do it with 600, it sweeps out
38:50everything that's to the left.
38:52In other words, the area of 900 I
38:55subtract the area of 600. Let's
38:57calculate the area of 600 first.
39:01Then it would be distri pun point nor
39:09and I'll take point N. From which 600
39:14comma
39:17the mean, comma the standard deviation,
39:21comma true.
39:23And that's it.
39:25Now I'm going to do it. Notice that here
39:28I calculate everything that is below
39:31600.
39:33Now I'm going to calculate everything
39:37that is below 900. So, again the
39:40function distri pun nor.n n of x which
39:43is 900 comma
39:47the mean, comma
39:52the standard deviation, and comma true.
39:58And notice that I can look at the
40:01function with f2 to analyze it.
40:04So we've already calculated everything
40:07that the area of 900. So, from the area
40:09of 900 I subtract the area of 600. Let's
40:12do it here.
40:15The area of 900 minus the area of 600.
40:19Yes? It's the same to me.
40:21And this is to calculate areas above.
40:27This would be the area above 600 and
40:29this would be the area above 900. Yes,
40:32we can calculate it.
40:36Remember that it's 1 minus the area on
40:39the left gives me the area on the right
40:42.
40:44If you add these two areas together, it
40:45will give you 100.
40:50The probability is one, right?
40:53It gave me one.
40:54If v, because it's the one in
40:56probability, represents 100%of the data
40:59.
41:00Let's remove that from there and
41:03calculate this other area.
41:05So, it would be one, which is the
41:08probability of the entire space, minus
41:10the calculated area gives me.
41:13So, the sum of these two again gives 1.
41:16Well, interpretation of this 40%.
41:21So I would say that 40%
41:25of the students
41:28have
41:33between 600
41:38and 900
41:41hours of work experience.
41:49Okay, ready.
41:50So, we already have it in...
41:55I don't know if you have any questions
41:58or anything to ask about the normal
42:01distribution.
42:07Okay, ready.
42:09So, let's go to the next approach, which
42:12is approach four, which has to do with
42:15the uniform distribution.
42:19Let's see, I think I can steal that one
42:21.
42:22Let's see, this one, this one, this one
42:23.
42:27They run and run and they don't let me.
42:31Control C. I'm going to paste it here.
42:36There it is. Now, we're going to enter
42:39values.
42:40Let's delete this too.
42:44Okay, let's delete all this,
42:48all this and that's it. Okay, let's
42:50calculate.
42:51A, what's going to be the minimum value
42:54? Notice that here we have our uniform
42:57distribution that goes, which is a
43:00rectangle, between the minimum which is
43:03A and the maximum which is B. The height
43:07of this rectangle is calculated as 1
43:10over B-A, which is this distance.
43:13B, I subtract A, I'm left with B-A, 1
43:16over B-A.
43:18Okay, now it says approach four, suppose
43:22that the expected salary of students is
43:25uniformly distributed between 1545
43:36and 5,559,697
43:42pesos per month.
43:45X the expected salary of a randomly
43:48selected student.
43:50Calculate the following probabilities.
43:53Okay, so since you already told me that
43:56it was uniformly distributed, that's why
43:59I brought the rectangle here.
44:02So, for that I need to find the minimum
44:05and the maximum.
44:07So, the minimum of this data. Mm, but I
44:11think that Ah, yes, I already did it.
44:13The minimum of this data.
44:18Let's see if F2 is okay.
44:21Done. Good.
44:24And we are going to calculate B, which
44:26is the maximum. Yes. Look, here is the
44:28minimum and here is B the maximum.
44:31So, the function is max of the data,
44:34that is, of the salary, the salaries
44:36that we have.
44:38Okay, we already have B-A
44:43is what we have, the length of the box.
44:45Do you see? So let's calculate B-A as a
44:48help.
44:54It will be the denominator of everything
44:55we are going to do here.
44:57Ready? So let's look.
45:00He says, "What is the probability that a
45:03student expects to receive a higher
45:06salary?
45:13We have values, note that my a is 1600.
45:18So,
46:27microphone. Hello, hello, hello, hello.
46:32Can you hear me?
46:34Yes, now.
46:35The audio went out for a moment.
46:39>> The audio went out. I don't know if you
46:42caught the last part.
46:48Or should we repeat it?
46:50You couldn't
46:52>> hear it, teacher.
46:54>> Okay, so let's review again. The minimum
46:57. Did we find it? Yes. How far?
47:02The minimum from there or further ahead
47:06.
47:12So, from the minimum. Okay, let's start
47:14the whole approach.
47:18So, to calculate the minimum, then we
47:21call it min
47:23of the data. We select the data
47:28and enter. We already reviewed F2. There
47:33are the data.
47:35So, we already have the minimum.
47:37According to our graph, B is the maximum
47:41. So, max
47:44of the data.
47:46I select the data and enter.
47:50And now we are going to calculate B-A,
47:53which is like the denominator that we
47:56have here.
47:57So, it would be B-A.
48:03What for? To get the probability of
48:06occurrence which is 1 over B-A.
48:08That's it.
48:10Now, then I'm going to erase this little
48:13piece that we made here.
48:18And that's it.
48:20So, the first exercise asked me, what is
48:22the probability that a student expects
48:25to receive a salary greater than 4
48:27million?
48:29So, the reference point is 4 million.
48:32Yes? Well, and I'm going to place it
48:35here in this rectangle.
48:38So, A is more or less 1600 would be here
48:41and B 5,500
48:46more or less and I'm going to place
48:484,000 ones, that is, it would be more or
48:50less around here.
48:56I'm just going to write it like this.
48:59And here it tells me greater, so it
49:02means that what I have to calculate is
49:04this, okay?
49:07So I'm going to do the whole process.
49:11So first I'm going to calculate the
49:13lesser, that is, I'm going to calculate
49:15this little piece and then I'm going to
49:17calculate the greater.
49:18So,
49:22to calculate this piece it would be 4
49:24million, which would be this,
49:29minus a which is the minimum.
49:34And that gave me this width from A to 4
49:38million.
49:40So now the height is B-A.
49:44I already have it there, I already
49:46calculated it.
49:48So, since we know that the blue
49:50rectangle is worth 100%of the data, I
49:53can do the following.
49:55This little blue piece was worth 61%
49:59or so. In other words, this little piece
50:02is going to be worth
50:0539%or so, a little bit less. So, I can
50:08do it by difference because the
50:10rectangle is 100%.
50:12Let's do it on that. So I say 100%
50:17minus what's on the left and that gives
50:21me this on the right. Yes?
50:24So, here it was. It says," The
50:26probability that a randomly selected
50:29student
50:32receives a salary greater than 4 million
50:35per month is 38.85%.
50:40Since I changed the data, that's why I
50:42changed that.
50:4438 point
50:4666%.
50:50This means that under the assumption of
50:53uniformly distributed salaries
50:58here, it changes the same because I no
51:01longer have that minimum, if I do not
51:04have a minimum of 1 million, 1,600, 43,
51:14again.
51:22And 5,000,
51:27485,037.
51:34Done.
51:38It says here that approximately 39
51:44students out of every 100 would have a
51:48salary expectation above 4 million pesos
51:52per month. Yes? In other words, it is
51:55like what I can predict through these
51:59values obtained. Yes. Okay, let's
52:02continue.
52:04It says, "Which one?" Let's take the
52:06graph to help us. Control C,
52:10control B. So it says, what is the
52:13probability that a student expects to
52:15receive a salary of less than 2,500?
52:21So, it means that I have to look for,
52:24remember that this was 1600 and this was
52:275,400, almost 5 million.
52:30In other words, I would say
52:32approximately that 2.5 million is here.
52:362.5 million is here and what I need to
52:39calculate is the lower part, which would
52:41be this.
52:43So, first I have to calculate the base.
52:46It tells you, let's see, the base is
52:50parentheses, it would be 2.5 2.5 million
52:54minus a, which would be this, -a.
52:59I close parentheses over
53:02B-A, which I already calculated.
53:06And that gives me the area below. So
53:08there I can do the interpretation and I
53:11say
53:1722.29%
53:22of students
53:26have salaries
53:30below 2.5
53:35million
53:42, that is, they are not, let's say, uh.
53:4822%is like more or less a quarter have
53:51salaries of 2 and a half million,
53:54meaning that students have quite good
53:56salaries.
53:59Well, here's another question that says
54:01, what is the probability that a student
54:06expects to receive a salary greater than
54:093,500?
54:11So, we write down 3,500.
54:15There we have it.
54:16I'll steal the little graph again to
54:19help me understand what I'm doing.
54:24So, remember that this is 1600 and this
54:285,000 is 485.
54:32So, more or less 3.5 million will be a
54:35little bit more over here.
54:39And here I say it's 3.5 million.
54:44So the area calculation says that those
54:47who receive more than 3.5 million, that
54:49is, it's this one.
54:51So, I'm going to calculate this length
54:55first.
54:57So I say, the length is going to be, uh
55:01, from where?
55:03From B, right?
55:05I subtract 3.5 million from B. So I say
55:07B is the maximum.
55:11The maximum
55:14-3.5 million.
55:18And that's going to go over B-A, which
55:21is this whole dimension, right?
55:25So over B-A, which is this
55:29to determine the probability, right?
55:32I mean, you give it everything and you
55:34say, "But I just want this little bit."
55:36So it says, "Oh, that's 51%."
55:40So, since this is the smallest, right?
55:44So there, what did I do? Look, look what
55:47we did.
55:48Here I did it wrong because I wanted to
55:50calculate that directly. I'm going to
55:53delete it because we didn't have the a
55:56in front of what we have here.
55:59Here what we have is, if I write in
56:03probability, it's the probability that
56:07the salary is less than
56:123.5 million.
56:14I'm going to write it like this.
56:20And I'm going to paste this so I don't
56:22have to copy the number again.
56:25It's the largest. Yes. So, I was doing
56:27the largest directly and no, no, because
56:30we're going to go off the template.
56:33So, let's calculate the smallest first.
56:36So, the smallest, notice that it's going
56:38to be this little piece, this that's not
56:40shaded. So, let's calculate it on top of
56:43it.
56:44So, how do we calculate that? So, it
56:47would be 3.5 million
56:49minus A, right? Because it goes from A
56:53to 3.5 million.
56:55So that's calculated as 3.5 million
56:58minus A, which is this over B-A.
57:04That's it. would be the area in blue.
57:08Now I'm going to do the shaded area.
57:12So I did it before, it should be the
57:14same for me. So it's 100
57:18%minus 48.
57:23If you see, it was same for us. I did it
57:25with another process, but it's the same
57:27.
57:28See?
57:29So I have to do the interpretation of
57:32the higher one, that is, of what's in
57:35pink.
57:37So I'll say that
57:4151.67, 67%
57:47of students
57:54receive
57:57a salary
58:00higher
58:02than
58:063.5 million
58:09pesos
58:11per month.
58:15Okay, so notice that there are different
58:17ways to do the interpretations. Let's
58:20take the graph, which helps us a lot.
58:23Notice that I could have done the
58:26exercise directly and it gives me the
58:29answer, but then we have to respect the
58:32template so that it's more organized.
58:35Okay.
58:36Well, let's remove this.
58:39Let's remove this. So, let's leave it
58:41there. It says, "What is the probability
58:44that a student expects to receive a
58:46salary between 2 million
58:51and 3 million pesos?"
58:53So, let's graph it first, let's graph it
58:56so that it's clearer.
58:59Between two, so the minimum wage was
59:041600, that is, it's closer here.
59:11Between 2 million and 3 million
59:16more or less over here.
59:19So, what I'm going to calculate is this
59:21.
59:22Ready?
59:23So, let's calculate.
59:25Let's do, remember that capital f is the
59:28cumulative. So, this would mean that it
59:31is the probability that x or the salary
59:34is less than 2 million. Okay? I'm going
59:38to leave it like this.
59:41And this will be the probability that x
59:44is less than, uh, sorry, I got it wrong
59:48above.
59:49Less than two is this one, it got it
59:51wrong.
59:52Less and the next one was greater.
59:57Well, because this is what's on the left
59:58and this is what's on the right.
1:00:01Let's calculate.
1:00:03So, uh, I'm going to calculate those who
1:00:06have 2 million, that is, I'm going to
1:00:09calculate this.
1:00:12So, let's calculate that. Draw. Wait a
1:00:16second and see if I can change the color
1:00:19. No, it won't let me.
1:00:22It let me. No,
1:00:25draw.
1:00:27No. Oh, yeah, I know why.
1:00:31Now, yes.
1:00:34Let's color it so you can understand a
1:00:36little bit better. Yes.
1:00:39So, for now, I'm going to calculate this
1:00:41little piece.
1:00:45That piece is going to match this.
1:00:49Yes?
1:00:51I mean, that little piece.
1:00:54So you understand what we're going to do
1:00:56later.
1:00:56So it's going to be,
1:01:01remember that if I'm going to calculate
1:01:03that, then it would be 2 million-a,
1:01:09which is this, I close and divide by b-a
1:01:13, which is this, and now I have the area
1:01:18in yellow.
1:01:21If I want to calculate all the students
1:01:23who earn more than 2 million, I would do
1:01:26this.
1:01:27Yes.
1:01:28So it would be 100%of the data. There it
1:01:33handles other probabilities. So it would
1:01:35be one minus this and it tells me that
1:01:38the majority will earn more than 2
1:01:41million.
1:01:4490%earn more than 2,000,000 and 9.3%
1:01:48earn less than 2 million.
1:01:53Okay, That's what we can deduce for now
1:01:55.
1:01:57I could copy another table here so we
1:01:59can...Oh, but it got damaged.
1:02:02Control Z. Let's keep it quiet.
1:02:06And we'll move a little bit over here.
1:02:08Maybe.
1:02:10Let's see. And we're going to do the
1:02:13same thing, but for 3,000.
1:02:16So, that means I'm going to calculate
1:02:18the black and white and that's it.
1:02:22Calculate all those who earn less than 3
1:02:24million.
1:02:26So, it would be 3 million
1:02:30A, right? Because that's all that's here
1:02:32.
1:02:33-A, that is, minus the minimum divided
1:02:37by B-A.
1:02:39There you have it.
1:02:43And I'm going to calculate what those
1:02:45who earn more than 3 million pesos are,
1:02:48so it would be 1 minus those who earn
1:02:50less than 3 million.
1:02:54So, that means that 35%earn less than 3
1:02:59million and 64%earn more than 3 million
1:03:03.
1:03:04But I need to calculate those who earn
1:03:08between
1:03:11two and three. So, pay attention to this
1:03:14little graph between two and three. So,
1:03:17I've already calculated those who earn 2
1:03:20million, which are these.
1:03:25And these are the ones who earn less
1:03:27than 3 million.
1:03:29Yes, these are the ones who earn less
1:03:32than 2 million. These are the ones who
1:03:34earn more than 3,000.
1:03:36So, but I need between two and three.
1:03:40So it would mean that from those with
1:03:423,000, all those who earn less than
1:03:443,000,000, I'm going to subtract
1:03:46everything that earns less than 2,000
1:03:49ones and those who earn between 2 and 3
1:03:51million will remain.
1:03:54Yes? And then I interpret. So I say, 26
1:03:58%
1:04:01of students
1:04:06earn
1:04:09between or have a salary better than
1:04:182 million
1:04:25to 3 million pesos per month, that
1:04:31is, well
1:04:36, that's
1:04:40not many, right?
1:04:41So, it means that there is a good
1:04:43proportion of students who earn more
1:04:46than 3,000.
1:04:48Well, let's look at the next one.
1:04:52It says, "What is the probability that a
1:04:54student expects to receive a salary of
1:04:57less than 3 million?"
1:04:59So, reference point 3 million.
1:05:03Now, this one, what is it going to do?
1:05:08This one, we are going to calculate
1:05:14the salary of three more or less, it's
1:05:16over here.
1:05:19And I'm going to calculate 3 million,
1:05:22those who earn less than 3,000 ones,
1:05:24that is, this little box.
1:05:28So that's going to be
1:05:323 million
1:05:35minus the minimum wage
1:05:40divided by b-and there you have it, I
1:05:44have it there. So, now I can interpret
1:05:47and say that
1:05:5035.31%
1:05:53of students
1:05:57receive
1:06:01a monthly
1:06:05salary of less
1:06:11than 3 million pesos.
1:06:18There you have it.
1:06:20We have the last one left.
1:06:22What is the probability that a student
1:06:25expects to receive between 2,800 and
1:06:284,200?
1:06:30So, let's paste our graph, which is very
1:06:32helpful.
1:06:35So, we go between 2,800
1:06:43and 4,200.
1:06:454,200.3.
1:06:48There you have it.
1:06:49So, we are going to calculate only what
1:06:51is necessary here.
1:06:53So, 4,000 ones, uh, according to what we
1:06:56have seen, 4, uh. 2,800 is almost here,
1:07:023
1:07:032,800
1:07:07and 4200 is a little bit more here,
1:07:124,200.
1:07:15So, what I'm going to calculate is these
1:07:18two, okay? The area between those two.
1:07:21So, I'm going to calculate all those who
1:07:23earn less than 4,200, all those who earn
1:07:26less than 2,800.
1:07:30But first, I'm going to draw it in
1:07:31another color.
1:07:33Let's see, another color.
1:07:36Green, this little piece,
1:07:41because remember that we're going to
1:07:44calculate those who earn less than 2800.
1:07:47So, if we're going to calculate those
1:07:49who earn less than 800, they're the ones
1:07:52in this green.
1:07:55So it would be
1:08:00from 800 and we subtract it because we
1:08:03only go up to a, not from zero to a.
1:08:09We close it by dividing by b to a
1:08:13and now I have the area in green.
1:08:16Now I'm going to calculate green
1:08:20and yellow. In other words, that means
1:08:23that I'm going to calculate all those
1:08:25who earn between 1580
1:08:29and 1580. Notice that it's the minimum,
1:08:33right?
1:08:35643.
1:08:38I'm going to calculate between 643
1:08:41up to
1:08:44where I am up to 4,200 all of this.
1:08:49So that means I could operate these two
1:08:52, these two little pieces, right?
1:08:55So, I'm going to calculate all those who
1:08:59earn between the minimum and 4,200, that
1:09:02is, this.
1:09:05So it would be 4,200
1:09:08minus the minimum because I don't have
1:09:11values here minus the minimum which is
1:09:15A.
1:09:17And I divide that into b-a.
1:09:23I don't know what happened. Well, let's
1:09:24look.
1:09:26Oh, there was a symbol there over B-A.
1:09:32B-A is here and it was.
1:09:37So, there's the green area, but remember
1:09:41that this is both the green and the
1:09:45yellow.
1:09:47Yes, but they only ask me for the yellow
1:09:50.
1:09:52So, what are we going to do?
1:09:55So we're going to take away the green
1:09:56from all the green and yellow, which is
1:09:58all of this,
1:10:03and it gives me 36%.
1:10:05Now I can do the interpretation and say
1:10:08that
1:10:11hm 36%
1:10:15of the students
1:10:20have monthly salaries
1:10:26between
1:10:322,800.
1:10:41and 4 million 200.
1:10:49Ready. There were some questions.
1:10:53So it is important to note that since
1:10:56one and three are discrete distributions
1:11:00, one is going to compare what
1:11:03differences there were when calculating
1:11:07the same thing in approach one of, let's
1:11:11say, in section A of approaches one and
1:11:15three. If they were the same, if not, if
1:11:18they are not similar, why would that be
1:11:21?
1:11:22And those are the conclusions, right?
1:11:26And proposals, well, to say that, for
1:11:28example, since one distribution
1:11:30approaches another, then we can always
1:11:32use that distribution when such a little
1:11:34thing happens.
1:11:36Right?
1:11:37And here is the comparison of the
1:11:40approaches of the continuous
1:11:43distributions. So, what happened?
1:11:46For example, if I analyze approach A
1:11:51in the two distributions, what happened
1:11:53? Then I compare B in the two
1:11:56distributions, in the normal and in the
1:11:59constant, and see if there are
1:12:01similarities or differences.
1:12:04We say, well, since they are different,
1:12:07one always has the same probability, but
1:12:10the other varies because the
1:12:12probabilities change because it is
1:12:14bell-shaped, well.
1:12:16The probabilities are different. Right?
1:12:19So, we have to look at those
1:12:21interpretations so that we can make them
1:12:24consistently comparing each case, each
1:12:27statement, right?
1:12:30So, statement A of each distribution,
1:12:33the normal and the continuous uniform.
1:12:37B, we look at both, C and we see that
1:12:40definitely in some distributions they
1:12:43are similar and in others they differ
1:12:47completely. So, why is it that they
1:12:49differ? Okay?
1:12:51And here is the infographic that will be
1:12:53the activity that we are going to do
1:12:56next week. So, pay close attention to
1:12:58that.
1:12:59Remember to fill this out, you never do
1:13:01.
1:13:02Sometimes you don't fill out the cover
1:13:04page, so the cover page is also
1:13:05important, you have to fill it out.
1:13:09So, remember that this week it is
1:13:13individual work, but next week we will
1:13:17start to connect and show results from
1:13:21the entire group.
1:13:24Yes. In other words, the moderator
1:13:27should ask each partner to start
1:13:30condensing the results. Yes.
1:13:34And provide feedback, review among all
1:13:36of you what went wrong, what went right
1:13:39, and why.
1:13:41And when we have everything, That's when
1:13:43we can draw conclusions, right?
1:13:46We can't do them before, which is
1:13:48exercise three and four.
1:13:50Okay?
1:13:52Well, thank you very much for
1:13:54participating.

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