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Problem 5 Suppose X and Y are independent random variables following Uniformfo,1]. Let Z = (X +Y)/2 Calculate the cumulative density of Z Calculate the density o Z_...

Question

Problem 5 Suppose X and Y are independent random variables following Uniformfo,1]. Let Z = (X +Y)/2 Calculate the cumulative density of Z Calculate the density o Z_

Problem 5 Suppose X and Y are independent random variables following Uniformfo,1]. Let Z = (X +Y)/2 Calculate the cumulative density of Z Calculate the density o Z_



Answers

Suppose that $X$ and $Y$ are independent random variables with probability densities and $$ g(x)=\left\{\begin{array}{ll} \frac{24}{x^{4}}, & x>2 \\ 0, & \text { elsewhere } \end{array}\right. $$ and $$ h(y)=\left\{\begin{array}{ll} 2 y, & 0<y<1 \\ 0, & \text { elsewhere } \end{array}\right. $$ Find the expected value of $Z=X Y$.

Well that's probably been doing the following joint distribution. Now, the first thing we invite to find is the value of K and find the value of day. We know that the triple integral of K X y squared Z Must be equal to one. In order for us to a probability distribution. No extras from 0 to 1. Why goes from 0 to 1 and z It goes from 0 to 2. This means we have K times the integral from 0-2, brazil is easy. The integral from 0 to 1 of y squared. Dy the admiral From 0 to 1 of X dx Is equal to one. Now the integral From 0 to 2 of Z. Z is equal to The integral from 0 to 1 of why squared is one third And the integral from 0 to 1 Of that is 1/2. Yeah. Mhm. Mhm Two and one half canceled. So this tells us that K over three is one. Yeah, That's OK. Is equal to three. Yeah. Some things are joint distribution is really F of X. Y. Z. Mhm. This three X Y squared Z. With X&Y. Between zero and 1 Lindsey. Between zero and 2. That it helps us on B. Because one of the probability That acts as less than 1/4, Why is greater than 1/2 And Z is between one and 2. So finding this probability we want to integrate the triple integral of three at Weiss Birds. E Z goes from 1 to 2, Hind rose from 1/2 to 1 because it has an upper bound at one. Then Ash goes from 0 to 1/4 because it has a lower bound at zero. So this is equal to three times the integral From 0 to 1 4th versus the arts Times. The integral from 1/2 to 1 of Y squared dy Yeah, I'm gonna go from 1 to 2 of Z dizzy. So this is three times now. If we end great From 01 4th of X. Dx We get 1/32. If we integrate, why squared From 1/2 to 1, What do you 7/24? And then if we integrate Z DZ from one to We get three hands and then we multiply all these together. Mhm. This is 21/5, 12. Yeah.

This question we have to find out the density function of you which is equal to y once girl less. Why do scare where violent and vital? And this are independent. Standard, normal random variables. It has been shown that the distribution of both. Violence care and why do square this guy is square With one degree of freedom? Does both have MGF means more money generating function empty? That is equal to one minus two T. Raised of ar minus one by two. Where he is less than one by two. As we know that by one square and by two square have MGF. As given lead. U. Is equal to even by one square. Yes. A two x 2 square. Here. Everyone and eat were the constant. So the engineer for you mut which is equal to expectation of E. U. T. Put the value of you. It becomes a one by one scare. That's a two by two scared T. Now it becomes he raised apart even D. By one square and expectation of E A two T by too scared. And this becomes and violence care a 20 mm. Why to scare his duty here? Anyone Which is equal to a two which is equal to what? So that mut equals two. Empty. Andy Which is equal to 1 -2. Deep Race two Part -1. Know that this is the MGF Of the exponential drin are variable with Vita is equal to two. So that the density function of you is f. U. You which is equal to one x 2. He raised the bar minus you buy two Where you is creator are equal to zero. So this is also a chi square. Guys were distribution with degree of freedom to thank you.


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