Problem Consider tke following joint probabilities for the two Tnahl-e Xand %Find the marginal probability distribution of erph Show vour calculations_ Find the con...

Consider the situation of Review Exercise 3.75 . But suppose the joint distribution of the two proportions is given by $$ f\left(x_{1}, x_{2}\right)=\left\{\begin{array}{ll} 6 x_{2}, & 0<x_{2}<x_{1}<1, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Give the marginal distribution $f_{X_{1}}\left(x_{1}\right)$ of the proportion $X_{1}$ and verify that it is a valid density function. (b) What is the probability that proportion $X_{2}$ is less than $0.5,$ given that $X_{1}$ is $0.7 ?$

Yeah, that's probably been given the table and we would like to use this table in order to find the marginal distributions for both acts and why? We'll just begin by finding the marginal distribution franks Notice that actually takes on two different values. X takes on two and 4. So this means the death perfect is going to be a piecewise function With two different values Only taking on the values of two and 4. In order to find the probabilities, add up everything in the to column for eps. So add up all those different probabilities. I appoint one Plus .2 plus .1 0.1 plus 0.20 point one. That was everything in the X equals two columns. And adding those all together gives us 0.4. So the probability access to 0.4 now we do a similar thing for four. We just add up everything in the four column and when we add those all together It gives us 0.6. And so the probability that X is four 0.6. And so this is our marginal distribution tracks now for why we do the exact same thing except now we'll be adding up columns. Mr I'm sorry. Now we'll be adding up rows rather than columns. Yeah. So why it takes on the values of 1, 3 and 5? Mhm. Okay. And so let's add up everything in the one row for why To everything in the one rule for 3.1 plus .15 .25. So the probability wise one is 0.25, Okay .2 plus .30 means that the probability of Y is three is 0.50. And then lastly adding everything up in the five column for why Does this .25? And so here's our marginal distribution for what?

Well this problem we've been given the following probability distribution function and what we would like to find is the expected value. Okay. Mhm. And so um you select which is the expected value of it in general is equal to the integral from negative infinity to infinity of X times. Yeah, alphabets dear. Now here, since we've been given the joint distribution function, you know that are F sub X. Here we're coming to following the integral from negative infinity to infinity of the joint distribution function. The why? Where we integrate out the wine. Now you notice that this is defined only on the circle from X squared plus Y squared mean less than or equal to a square. And so this is going to become the integral. No, I'm gonna change everything here in the Polar X and poehler's are close in theater. We have times the integral from negative infinity to infamy of F of X, Y, X Y is one over pi square. Those are all constants. And so we don't need to change anything there. But since we went to polar, dy dx becomes our D R. D. Theta. Since we're going around a full circle, that fate is going to go from 0 to 2 pi since it's a circle of radius A R is going to go from zero. It's A. And so we just changed our bounds here. So to rearrange these interviews here, one over pi square can be brought up front. Let me have the interval from 0 to 2 pi of the co sin of theta. D theta. Then we have the integral from 0 to a. R. Times R R squared. Do you are mm This is one over pi a squared times the sine of theta Evaluated from 0-0. It's A two Pi. Mhm. R squared is one third. Are cute. Only take the anti Dreyfus. That's one third R cubed From R. zero. Today. Okay. We know that the sine of theta is zero at two pi and zero. And so it's just 20 And so both terms here go to zero. So this whole thing goes to zero. So we have an expected value of zero.

Yeah, that's probably been given the following joint distribution function. And we'd like to find first the expected value of X. Y square this is going to be the double some of X Y squared times the probability events. So we're going to some across our X values and some across are my values. And then and then together now whenever access to and why is one we have a probability of 10.15 as we have two times one squared because y squared times 0.15 Mhm. When access four. And why is one of probabilities 10.10 four times one squared times 10.10. Okay. Oh and why is three and access to our probabilities 30.25? That's too times three squared time 0.25 And you remember I'm swearing all my wife values because we had a wife square here Xs four and y is three. It's another 30.25 Yeah. When access to and why is five are probability is 50.5 point 15. And when access four and why is five? Probabilities 0.10 mm. And so we're just going to add all these together in order to get our probability student I mean are expected value. And what I'm going to do this gives us the value of 74 0.35 Uh huh. Now we also want to find the expected value of vaccine. The expected value of y you know, the expected value of X means we need to some across all the X values and multiplied by their probabilities. Right. And so this is two times the probability of two plus four times the probability of four. Well, the probability of two, if we add up everything in the to column for X is 0.55 If we add up everything in the four column for X. 2.45 and then we have these two together and this gives us 2.9. And so the expected value of ETS, there's 2.9 and then do the same thing for why? So this is the Sun across the Why? In terms of probability of white. Now, if we add up everything in the one row for why? It's probability is 10.25 If we add up everything in the three column three role for why it's probabilities 30.5 and the probability of 5.25 And so now in order to find this expected value, we're just trying to add all this together and when we do this joseph, the expected value of Why is three.

In this problem, it's required to find the value off New X and you Why? Let's start by New X. Your fax is integration off our boundaries from 0 to 1 for X and Y six exports three to play. Why the X do you want will be equal integration from 0 to 1 three over toe? Why do you, Roy, which is equal to 3/4 and for m y or mule boy? Thesis is mule Milton knew why so Integration from 0 to 1 seeks experts toe Why poor toe g x d y all these years? Thanks. So it will be integration from 0 to 1 two boy pose to do y which is equal to over three. So me, why is two or 3 a.m. u X is 3/4. Thank you.