a-class-has-10-mathematics-majors-6-computer-science-majors-and-4-statistics-majors-two-of-these-stu

Question

A class has 10 mathematics majors, 6 computer science majors, and 4 statistics majors. Two of
these students are randomly selected to make a presentation. Let $X$ be the number of mathematics
majors and let $Y$ be the number of computer science majors chosen.
(a) Determine the joint probability mass function $p(x, y) .$ This generalizes the hypergeometric
distribution studied in Sect. $2.6 .$ Give the jobability table showing all nine values, of
which three should be $0 .$
(b) Determine the marginal probability mass functions by summing numerically. How could these be obtained directly? [Hint: What type of rv is $X ? ]$
(c) Determine the conditional probability mass function of $Y$ given $X=x$ for $x=0,1,2 .$ Com
pare with the $h(y ; 2-x, 6,10)$ distribution. Intuitively, why should this work?
(d) Are $X$ and $Y$ independent? Explain.
Determine $E(Y | X=x), x=0,1,2 .$ Do this numerically and then compare with the use oo
the formula for the hyper geometric mean, using the hyper  geometric distribution given in
(e) Determine $E(Y | X=x), x=0,1,2 .$ Do this numerically and then compare with the use of
the formula for the hypergeometric mean, using the hypergeometric distribution given in
part (c).(f) Determine $\operatorname{Var}(Y | X=x), x=0,1,2 .$ Do this numerically and then compare with the use of
the formula for the hypergeometric variance, using the hypergeometric distribution given in
part (c).

Bryan Meares · Answer

Petey at Florida State University, has 14 statistics. Class is scheduled for a summer 2013 terms. One class of space available for 30 students. Eight classes have space for 60 students. One class have space for 70 students, and four classes have space for 100 students. The question A What is the average class size? Assuming each class is filled to capacity? Because in this problem were interested in the class size, we&#x27;re gonna let our random variable X be represented with 30 60 7100 to represent the students in each class. Because we have 14 classes, we can represent the probabilities of each of those students. Class size is to be one at a 14 for 30 ate at a 14 for 60 one out of 14 for 70 and four out of 14 for 100. So the question is saying, What is the problem? What does the average class size no average mor mu is found by adding together, each x value multiplied with probability. So we have our 30 multiplied by its one and a 14 60 multiplied by a 14 70 multiplied. Both won at a 14 and 100 times four out of 14. We can add together these four products and get an average class size 0 70 be spaces available for 980 students. Suppose that each class is filled to capacity and select a statistic student at random. Let the random variable hex equal the size of the student&#x27;s class to find the pdf Breck. So because we&#x27;ve already found our X values and our probabilities of those X values, we&#x27;re just gonna put those in the table to create their probability distribution we never expect. Used for 30 60 70 and 100. The probability of 30. It was one of 14. The 60 814 70 was one of 14 and 100. It was for out of port. You know that this is legitimate probability distribution is each probability 01 and then some of the four probabilities is one. See, find the mean of X, the mean of X. It&#x27;s the same exact thing is you&#x27;re expected value. So for this problem, we&#x27;re doing the same thing We didn&#x27;t, eh? We know that the means is 70 deep. Find the standard deviation. Our standard deviation is found. It was a pretty long formula, but it&#x27;s not that complex square root of the summation of the X value, minus the mean squared times. The probability because we have four X values. What this is gonna look like. It&#x27;s 30 modest, 70 squared multiplied by the probability of 30 which was one of 14 because that 70 it&#x27;s the same mean for every single one of these at 70 LaRue be repeated in all four for the next 60 minus 70 squared. Stop the probability of 60 touches 8 14 at that to 79 70. Weird, I&#x27;m the one about a 14. Lastly, at 100 minus 70 cleared four out of 14. Then we have to remember to include a radical drama radical at the end. Make sure it&#x27;s long enough. You can type this in. We can find that are standard deviation. 20 0.7

Robin Corrigan · Answer

in this question, a bunch of random experiments are described and were asked to identify which ones would have a random variable that is a binomial distribution and, if so, where as to identify the parameters in and P. And so to do this problem, I&#x27;ve just summarized sort of, the four criteria from the textbook, um, that are used to determine if it&#x27;s a binomial random variable or a binomial experiment. And just recall that if X is a random of binomial randy random variable, we denote it like this. Where in is the number of trials and P is the probability of success in each trial? Moving on to Part a were described X is the number of fours in 10 rolls of a fair die. So we know that there&#x27;s going to be 10 rolls of a fair die. Success is getting a four failure is not getting for and the trials they&#x27;re independent. There&#x27;s no roll of the die effects, a different roll of the die. So P is constant, which is always one out of six. And so yes, X does have a binomial distribution, and we could write it like this for B. It&#x27;s the number of multiple choice questions a student gets right on a 40 question test. When there are four choices. Reach question and the student is completely guessing with the 40 question test of their 40 trials. Each trial has success or failure. That&#x27;s right, Student gets it right. It&#x27;s a success. The trouser independent. That is true because the student is just completely guessing for each question. So no answer. Getting one writer wrong does not influence getting another question right around and P is constant. It is because the student again is just guessing, So they have a one out of four chance of getting each question right. So we can say that X does follow a binomial distribution and an is equal to 40 and P is equal to 1/4 now, for questions, see were said, we&#x27;re told that it&#x27;s the same as part be the same scenario, except that half of the questions have four choices and the other half have three. So going through their criteria, we do have an fix Trials 40. Each trial has success of fail or failure that&#x27;s right to be or failure the trouser independent, which is true the student is just guessing. So having the right answer wrong answer, for one question will not affect the outcome. For another question and P is constant. This is not the case here because if half the questions have four choices and half have three, the probability of guessing correct is one out of four further questions, with four and one out of three for the questions with three. So that is not met. So we would say no for C for D were given the number of women in a random sample of eight students from a class comprising 20 women and 15 men. For this one, we can say that the trials are not independent. And the reason why is because we air sampling from this class without replacement. So if if the first if, on the 1st 2 or three trials ah, women are selected or men are selected then on the subsequent trials, the probabilities depend on what happened in the previous trials. So for that reason alone, weaken, say no to D. Moving on to E. X is the total weight of 15 randomly selected apples. So we have 15 trials, except that we do not for each trial have something to outcomes defined a success or failure. We&#x27;re only adding the weights of all the apples, so we do not have a dichotomous outcome. So for E, we can also say no. And finally, F is the number of apples out of a random sample of 15 that way more than 150 grams. So we know it&#x27;s going to be 15 trails, so that&#x27;s met. Each trail has success or failure that&#x27;s met. We would call a success weighing more than 150 grams, the trouser independent. So it&#x27;s a random sample of apples. So the success for one apple was not affecting the success for other apples that&#x27;s met. And we can say that P is constant because there&#x27;s some proportion of apple is that way more than 150 grams. It&#x27;s a few sample randomly from the population of apples. That proportion P is constant, so therefore we can say that excessive binomial random variable number of trials is 15 and we don&#x27;t know the probability success, but we know that there is some P Valley for that

Pawan Yadav · Answer

Hi. So we have a question here to calculate the probability. So data given us we have a number of tablets. That is 20. So we&#x27;ll take this as capital in, and we&#x27;re displaying every radio with his number off. Six. So that is a what is the number we&#x27;re displaying? Okay, then we have that out off this. 20 tablets, two slots tablets are 12 on That is what we have to display. So this is a success rate, or we can call. This is our success represented by capital M and number of tablets with one slot will be 20 minus 12. So that is eight. So we can like this is and minus m just eight and were given random variable. Okay, so first of all, we have to find out what type of distribution it is. So if you look into the data, it is given that you have ah, population and out off that we&#x27;re selecting some off it where each off them individually have a constant probability. So such kind of distribution is Caldas Hyper geometric Probability. Okay, so what type of distribution? He&#x27;s hyper Giammetti. Okay, Now the parameters that are here is capital and which represent total number already mentioned. Okay, out. Off which we&#x27;re selecting. Six. So that is a selection number. We presented the best small lane, then the success rate or the number off success. So success represented by Capital M his 12. And the failure one is in minus one, So failure is n minus. Um, she&#x27;s 20 minus 12. It. Okay, so these are the parameters. Now, this is a part of a solution for part B. We have to calculate probability off X equals 22 to calculate probability off X equals two to we have to find What is the chances off selecting two slot tablet to display? Okay. Okay. So first I&#x27;ll write. What is a total number off tablets? We haven&#x27;t out off that. How many we are selecting. So we have 20 tablets and we are selecting How many? Six. Okay, so 20 C six serie present permit a combination similarly in numerator will have out off this. How many? We have success. Okay, so we have we have to have to success. That is, we have to have two tablets with two slots. Okay, so we have 12 tablets off that kind and out off that we&#x27;re selecting two. And the remaining tablet will be selected from the failure one. So number off failure tablets is eight and out of that we&#x27;re selecting four time. It&#x27;s because we have already selected to. If you calculate this value, we get it around 0.119 which can be returned at zero point 12 Okay, now, next probabilities Probability off X less than equals toe to and that is equals to probability. Off X equals to zero plus probability Off X equals to one plus probability off X equals two to great. So we just have to apply the same formula three times. So we have to select zero tablets with two slots. So we have total 12 tablets and we&#x27;re selecting zero. And since we&#x27;re selecting zero tablets so the other six will be selected from the tablets which is one slot so out off a tablet, we&#x27;re selecting six tablets and here we&#x27;ll have the same time. Similarly, for X equals to one that will be too well. See one 81. Sorry. Eight C five. Since we have selected one from two slows tablet, you have to select five from one slot tablet, and here it will remain the same. And be off X equals 22 we have already calculated. So we can directly right that when you hear. Okay, So, on calculating this value, we get probability off excellence than equals Toe to is 0.144 So this was plus solution. It&#x27;s the second and the third one is probability off X greater than the question too, since we have already calculated probability off X less than equals two to where now we have to calculate X greater than equals toe. We can inward this by applying a property off probability. Complimentary probability that is one minus probability off eggs less than two. Since we know probability off a is one minus probability off a compliment. Okay, so this becomes one minus X less than two. So that is probability. Off X equals to zero plus probability off X equals two. What? Now we can calculate this value that gives us one minus what we have calculated there. So I&#x27;ll directly right down. So it&#x27;s zero point 87 three. Okay, so that&#x27;s a value off part B off our solution. Next we have to calculate me and standard deviation to calculate. Mean so part C. I mean and standard deviation. Standard deviation. Okay, so it is e off X is 10 times m or in and standard deviation. It would be rude, off N m or M times and minus and or n minus one times one minus Mbai. And so we just have to substitute the values and calculated. So when you off N is six Ali off m s 12 when you off N is 20 and as d is too tough again, this is six inches to 12 or 20 and it&#x27;s again 20 minus small. Any six or 20 minus one times one minus m is 12 or N, which is 20. So on calculating this value we get So it is 3.6 and standard division is one point. Seriously, that sounds self foresee. Thank you

Heena Haldankar · Answer

the margin distribution of X is given to us in the questions, I will just read out that part. So the margin and distribution or ex. So I&#x27;m gonna draw that table here. So they have given us the values off X and, uh, zero. Then we have one next to stool three for. And, uh, the probability off each of these values is also norms, which is 0.1. Next. This 0.2. Then we have 0.3. Then wind blew five and wind one fight. Okay. Now, to start with the day after that, X over here represents the number off canon Digital camera sword during a particle of it by a certain store. And, uh, by what he represents the number of poor chases during this week who by an extended warranty. So to start with the first part what they&#x27;ve asked him the question. We can see that and where he was given a sport and the probability where sixth dimension that 60% of customers who purchased camera also where warranties, which means 60% is wind. Six over here as a provider beauty. So the condition probably D In this case, we can write what it asked. Well, access for depending on when bias to. So this is given by a probability off by equal Toto when X is equal to four and this into probability, off X is equal do for so well what you were going to use the binomial expansion formula. So that formula is given by N. C. X. Then we have the probabilities two eggs and one minus periods to end minus X. So this is the formula that whom do you use for the by normal expansion? So if I sold this part of a Sure, then I can start writing the values that this will be equal do for C two and, uh, why in six days to do then we have 0.4 trees to four minus stores toe and into the provider tee off X is equal to 40.15 So now when I substitute these values, then we get the value as 0.5184 So, as I said, we have taken four seater because we have to be a low and, uh, values of Iron X is equal to four. So based on that, we&#x27;ve applied this form like here and moving on to the next part. They&#x27;ve told us to find the joint probability Merapi off X is equal. Dwight. So for the part B, they&#x27;re told to find be off X is equal to y. So this means that we have to consider the all the values of X environment. They would have the same value where it will be from 0 to 4, and we multiply that with the probability off that particular value. So to start with the Barbie, I would say pee off X is equal to buy. We can ride this in detail as with the formula, and be off by a zero ex Tzeitel. And we take into probability off x zero, Right, Bless The next value is in water. One to be off. Why is one when X is one and do be off X is one then I have placed. Be off by is to annex is duke and so this is be off X is equal to do next we have the off. Why is three when excess three and into probability Off X is equal to a tree and lastly, the value your vies for when access for and into probability off actually will do for so the PM fs are already given to us also, uh, the by normal expansion formula that I apply So we know the value of P and and what? Their second directly a play on the formula for the 1st 1 is zero sees. You know we will have 0.63 straws. Edo on the 0.4 values also is 20 and into the sea off X equal to zero. Value is 00.1 Bless the next is one C one, then the air 0.6 days to one again point forward Big agrees to zero on in to execute. One probability is given us point for the next values to see toe we take 0.6 race toto point for the game grease to zero in the probability of X equals Togo This 0.3 plus we have three. See tree, then 30.6 Ray&#x27;s tow three when blindfolded again, raise to zero and into the probability of X equal to three years point through five class the lastest four c four, then 16 ways to fold. And this is point for the race to zero again and in tow. Creepy off X equal to four is given as 40.1 fight. So if I can see it all of these values I get this ice 0.1 the Force Thomas 0.1 plus the next is 0.1. Do place third is 0.108 less 0.54 plus 0.19443 adult. Off them the finance of what? This party getting a 0.40144 now moving on to the next part. They told us to find the joint p m f off except by So we want to find out the joint be America Off X anyway, in this case. So if I use the formula that is given to us they have said that be off X camera. I will be returning as be off. Why is he good wife? Where X is equal to any value X and we go into the publicly off X is equal to the value of X. So ask for this. If I apply the formula, then this will change to X right, the combination&#x27;s partier and then we have 0.6 is released away. Then I have point for will be released to X minus way and in to live a little B off X. So now, based on this, I&#x27;m going to formulate our table and calculate the values. So first, I&#x27;m calculating the value when bought our zero So the present I&#x27;m finding the value for pee off 00 in that case is changed to 00 Then I have 00.6 is raised to zero point food. And also these 20 run into P off zero X is equal to zero has given us 00.1. So if you simplify, you&#x27;re getting this value ass 0.1. Next is if I take suppose the value off 10 then this changes to a combination of an zero with 00.6 days to zero. When full is also this to no one on in tow. The probability is zero point Go for X is equal to one. So if you simplify it in this alley that you will get is 0.8 So I will continue calculating all of the values are up to when you reach be off for comma zero and your four comma zero, the value would be 0.384 for the, uh if I have to find the value off the other Randy&#x27;s in the table, someone who consider the table here and fill the values. But you calculate all of these values using the same formula in the same bar down someone who consider that table here. So I just resize the table first. Oh, yes, you could see I have taken the columns and the rules. So the columns are the exiles and those are the values. And based on this, I&#x27;m going to write down the values and the table that we&#x27;ve got soapy off 00 where here is the first value. Let me just put the value here. This is 0.1. Then we&#x27;ve got 0.8 and you got hurt. All of these values in a similar V then Now we will get the total value yours 0.48 then this one is 0.16 Then I have ah 0.384 moving further for 01 We do not need the value so that we may be able to move for them, and this will change to 10.1 toe. Then we have joined 144 The next value is our 0.72 Then I have ah, 0.23 04 Then for those I have ah, going to 108 Then I have again point ones you ate in the last. Really? Your is 0.51 heat. And, uh, this really here is 0.0 fi four. Okay, this is again 0.5 18 And the last 20 that I&#x27;m getting here is zero point 0194 So if, uh, zero here, if I had all the political values, I&#x27;m going to get the PM for fights. If I add vertical values, this value was point or who? For 784 then the second value. I&#x27;m gonna have your zero point 359 or four. And the third that I&#x27;m getting here is joined to six, 784 in the next. His boy nde 10 Why, he&#x27;d four. And then here. This is as it is, a 0.1944 And now, based on uh, the horizontal additions of the values I can get the b m for fix opium affects is, in fact, already given to us this 0.1 in the question to appoint toe 0.3. Then the next. His 0.25 And I have wind 15 such that additions or the war will be won. So from this table, they&#x27;ve also told us to find the marginal b m f off. Why so and what I&#x27;ve done the marginally be mm off. Why? So you&#x27;ve already got that from the above table? So why the leader from 0 to 4? So I have I and I&#x27;m gonna ride on the probability off. I hear first value zero. And for that the probability is 00.24 784 Then the next one. He was one so far that the value is 10.35 904 Then you have to sew. It is 9040.26784 The next is for three. We have 30.10584 And in the last is four. Where the value of the probability is 0.1944 So this is the marginal pmf off. Why

Heena Haldankar · Answer

All right, now in the question here they&#x27;re given is an expression for pdf off hyper exponential distribution. So the first, but just about is asking us to find the CDF for this particular pdf someone to write down the expression. We know that silly f and B Okay, we have to write this in terms of the variable. So he had the label is Lambda one. Then we have to. And then the fourth variable is be so well today this as, ah, integration off the given. Pdf from, say, zero West. So I want to write X as why we&#x27;re here for simplification and again the valuables or the parameters as Otis and we&#x27;re in debating this with Why So if I heard the expression that we have been given in the question don&#x27;t buy or healed. So this is integral zeroed wax and what was substituted The values we have be number one. Eat a stew minus lander one by plus the next borders one minus B into London too. It is to minus London to buy mint. Anyway, So now we will deliver this with respect. Oh, why soapy into Lambda One is a constant. I&#x27;m digging it outside. Insulation off, idiots to mine Islam that one by upon this is mine. Islam. No one says from zeroed works, so we can cancel on the one here. Okay. And further, this is one minus B into this Islam dental. And the indication of this will be eating is to minus Lambda Tau by On mine is Lambda Tau. Okay. And the limiters, You don&#x27;t ex, so I can cancel Lamberto also. So the value that I&#x27;m finally getting here is so this is minus B in the black. It now people, top element. We have meetings to mine. Islam, double necks minus Easiest was always One plus is one minus B on the bracket. We have Edie&#x27;s toe minus Lambda Tau X minus one, because it&#x27;s easiest to zero and this whole thing upon minus one. So if we give the minus science inside the bracket, the final time that I&#x27;m getting here now is be in the racket. One minus. It is to mine. Islam. The one X and this will be a plus one minus b into this minus. I&#x27;m giving the new military. So you have one minus. It is to minus Lambda Tau X. So this is the CDF for that? I have fun for this fabulous distribution. So therefore, toe rather finance that I can write down the CD of as this is a B is equal. We convert downtown&#x27;s off limits and they&#x27;re going to be zero for, say, excess lesson zero. And this will be be into bracket one minus. Used to mine is Lambda one X plus. We have one minus b into the bracket, one minus used to minus Lambda Tau X. This is going to be for X rated than or equal to zero. So we&#x27;re done with the part, eh? Not moving on to the part B The question tails me. Oh, that I need to, uh, use the values. Given the question be undone. Lambda Tau Tau actually find on the castle. The value of probability of X is greater than 00.1 So true. Start with upon me. I wonder I don&#x27;t the values that they have given to us. So we have been known that he is, you know, 0.5. Then we know that Landau one is 40 then. I also know that land it was 200. This is given it wasn&#x27;t a question. So he wants to find probability Off X is greater than you don&#x27;t mind? Yeah. The question see, is it is zero point 01 So this probably part I can write as one minus B off. This is X is less than equal to 0.1 So this case this will be now equals one minus. I can learn this as capital left off zero point 01 and finally this candidate and has one minus for finding this value, I can use the CD of dust that we&#x27;ve just calculated above. So, using that concept, I&#x27;m actually going toe find that value now. So substituting those values along with B and Lando Lando, this will change 2.5. The black it we have one minus is too minus 40 into this is 0.1 Get is the first part, and then I have bless. One minus 10.5 will give me 0.5 again. And, uh and I&#x27;m not gonna have one minus e stool minus 202. This is 0.1 So now if he has lived this entire expression, we&#x27;re going to get the final Answer this for me. We have one minus. This is, uh, wind. Five and in the bracket were using the calculator here. So opening a black it again. And this is one minus. Yeah, we have e race, too. This is 14 2 points. You no one is going to give me going for some minus point for And, uh then I have one Does This is minus 0.5 on the bracket. The they have one more dumb so that in the record, we have one minus. This is Edie. Stool. My stool. Yeah. So we&#x27;re getting the Internet&#x27;s 0.40 to eight. So what do I don&#x27;t find an answer Here is blind 40 go eight. So this is what we&#x27;re going to Probably the venue for the second part now moving on to the tall part. The question for the tour part, see, is that, uh if X has ah, the given if as a speedy of what does the EOE effects. So we&#x27;re supposed to find the expectation for the given, Pdf, You know, the affectation, the value that we have weaken Diogo. Oh, say let x one and x two Okay, B Oh, exponential landen valuables. But though, but I mean, does as London one and learned up to. So in this case, therefore, I can express X in terms off be into X one class. This is one minus being brew x two. So applying expectation to this expression and then therefore say that your ex is actually equal. We apply. Expectations only functions off. So he&#x27;s a constant women like that only we will apply expectation to x one Bless again. One minus people. Um, in the saddle. Me and we have life expectations. Rex too. No for this distribution, the expectation for X one will begin by one upon Lambda one. And expectation for extra is even by one upon them. That Sofus observe these values over. Sure, I can therefore find out the main expression for expectation. So it&#x27;s going to be off on land. Are one plus one minus B a born than that. So this is the answer for the expectation off this particular function. Now moving on to the next part Barbie. The question is that we&#x27;re supposed to find out now the expectation of X squared and, uh, they have given us an expression that expectation off X squared is drew upon land a square. So, using this, I&#x27;m going to find out why. Continues distribution. What is the expectation of X squared? So I can therefore say that now, by definition, for continuous distribution, expectation off X squared can build on it as integral 02 infinity. It&#x27;s actually minus infinity plus infinity. But we&#x27;re leaving zero to infinity because the question is existing for positive function. So this is X square into FX the X Okay, so then that is equal to integral d o to infinity. This is X squared in tow. FX is the pdf, which is given. It wasn&#x27;t a question. So I&#x27;m going to ride that years of spiel under one Andi eat is to minus No, no one X Okay. And then Plus we have one minus b. The lambda to it is to minus lambda Do X with the ex so we&#x27;ll hear the lander one and one minus freedom that were Constance and between X Square. And here&#x27;s to Tom. We&#x27;re actually supposed to apply the concept of expectation of X square, so I&#x27;m gonna take the lander one outside, and this is in Diggle zero to infinity X square. It is to mine Islam. The one ex, the ex less. The next part of what I have is one minus B into Lambda to immigrant, zero to infinity X squared and Dewey&#x27;s to mine is lama to expect the ex. So now this part, I can actually write as, uh the e off x squared for X one. And this becomes for extra. So I&#x27;m replacing the terms with B. Now. No one has situs. And this I&#x27;m representing as expectation off. This is excellent square. Okay. Plus, the next is one minus B in tow. Tow. And, uh, you can actually read at this whole function with removing the dorm off lambda, um, one in love that you&#x27;re because you want to give that as a part off expectation of excellence, Quinn and expectation Extra square. Also, that this actually will now change to pea off expectation off. Excellent square directly because, uh, well, I&#x27;m not gonna Lamberto apart off accidentally functions, and then we have Plus next, just one minus p into this is expectation. Off extra square. So this is what we&#x27;ve got as expectation off x square. But it has been given that this morning we have told you of X squared is true on land a square so I can substitute or use this concept to substitute the values here. So I can therefore write that now you have X squared will be actually quitting parameter for Explain Islam, dominant barometer for extra Lambda too. So this will change to be in tow upon Landau one square. Plus, this is one minus B into do upon lambda do square. So if I read it, this is the proper form. The expression that I&#x27;m now getting is to be upon Lando and square. Plus, this is twice of one minus B one time that&#x27;ll square. So this is what we&#x27;re going. The expression for you off x square, but moving further. You&#x27;re also supposed to find the expression for aliens off X. So we know that the formula for readings off X is given by expectation off X square. Minus the mean, which is you affects the whole square. So we performed. Boulevard is already so I&#x27;m going to write down. The value is here. This is to be by mandolin Square blessed rice off on minus B into them upon Lombardo Square and then you affects the whole Tequesta. This changes to minus bracket. We have be a born Landa one and this plus one minus B upon them. That too, the whole square. So this is what is the value for go billions? No, moving further. The body. The question states that we&#x27;re supposed to use the explanation institution to find the core radiance here. So I&#x27;m going to use the formula off the meaning standard deviation off exponential distribution to find the cove aliens. So the cove, Aidan&#x27;s candidate and as Sigma upon meal now for the exponential distribution sigma is given by one by Lambda upon mean is also given by one by land us If you simplify, this actually gives me one. So we can therefore control that the co variance for an exponential distribution is equal to one. Now, uh, therefore, I can use the part c and the party that we have proved about in this particular some toe find, Ah, the cool villians off the high board exponential distribution because we need to find it for this also. So now, uh, I know that the Corbin&#x27;s Sigma upon Mu can also Britain as doesn&#x27;t re underworld off. We have the evidence, which is to be a born land a one square places this twice off, one minus b a born London to square and this is minus. Extend the bracket. We have be off Orlando. One plus is one minus B a porn lambda through the whole square. And the whole thing is now divided by expected him Just be upon around the one plus, yeah, one minus p upon Lambda too. So I want to value this expression in such a way that we can try and simplify it in a better be So I&#x27;m I don&#x27;t think this is a normal expression under the rule cure, so we can actually write down the whole thing again. So this is Toby by number one squid. Plus we have twice off one minus B upon this Islam dark blue square and minus the Stanmore. Right. You&#x27;re as say, Nanda do be what I can I have to be. I like it like being number two. Bless. This is Lambda One into one minus B. The holding A born. You have longer one time that you and the square and said the bracket and the whole expression is again divided. This we&#x27;re taking it under the hood now. So the expression if this got into the triple hole square again and I&#x27;m cross my playing it. So we have be into Lambda Group plays. This is Lando one in the one minus b. The holding A born Landa one them that, too. And the whole racket square Because we have taken this undergo group. So if you see, I can actually split this into the numerator and the denominators and the expression that I am goingto finally. Now get is this is under cough. You have? Ah. If I cross multiplying the four stones, we will get Toby in tow. Randle square. Bless. This is, uh, two into one minus. B induces lambda one square. Okay. And the whole thing is divided by lambda. One square on dhe, lambda to square and the right inside water means only minus one. And under the road itself, the school thing is again divided by the actress expression that we have is being to Lambda do less land a one in the one minus p. No one square back close here and upon says Lambda. One square and a few square. So this is the whole expressions under the hood here, so we can cancel these two domes, Okay. And, uh, finally, the remaining dome that I&#x27;m getting here and we&#x27;ll read. I did. So I&#x27;m getting this. A strays off the black it I can read this a speed on that square. Plus, this is one minus B into lambda one square, the whole racket close upon. He would I have a feeling I&#x27;m going to bless. This is number one into one minus. It&#x27;s me. And here it is. Whole square again. Okay? And this is minus one as it is so devalued. This enough for the way we can say that Generally a probability, or like being one minus Peter present probabilities. So, probabilities we know it always lie between the 11 So max to max value the gravity can take is one. So if I see the value of this whole expression on an average or the numerator off is actually at the stairs large as the denominator, and we can therefore said the asshole fraction of actually going to be greater than or equal to one approximately so then, In that case, I&#x27;m actually now want to simply find this and say that signal upon you can Therefore, Britain, as is greater than equal, wondered off price off expression. I&#x27;m substituting by one minus one. So therefore, I conclude that sigma pawn you, I&#x27;m going to be greater than or equal to one. So this case, we can see that the obedience foreign hyperextension distribution will be greater than or equal to what? No, you didn&#x27;t get the part. We wanted the next part just the part f. It is talking about the old and distribution. So for the Orleans distribution, the formula again for cooling its remains the same majestic ma born mule. But for a Nolan&#x27;s distribution, the Sigma is actually given by Rudolph in upon Lambda, divided by mule. The mean is given by N divided by Lambda Soudan. Cancel lambda shirt and be give the expression now, which is actually if you cancel the roots also so we can therefore get the core visions. Why sure as yeah, cool aliens over here is not actually got one up on food. And so we&#x27;re done with all the bots here. And this is the final also

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