Consider the pair of random variables (X,Y). Suppose that marginally X ~ Binomial(2, ) and Y...
Suppose X and Y are independent random variables with Exponential(2) distribution (Section 6.3). We say X ~ Exponential(2) if its pdf is f(x) = -1/2 for x > 0.
Suppose that X and Y are jointly continuous random variables with joint probability density function f(x,y) = {12rºy, 1 0, 0<x<a, 0<y<1 otherwise i) Determine the constant a ii) Find P(0<x<0.5, O Y<0.25) HE) Find the marginal PDFs fex) and y) iv) Find the expected value of X and Y. Le. E(X) and E(Y) v) Are X and Y independent? Justify your answer.
2. Suppose X and Y are continuous random variables with joint density function f(x, y) = 1x2 ye-xy for 1 < x < 2 and 0 < y < oo otherwise a. Calculate the (marginal) densities of X and Y. b. Calculate E[X] and E[Y]. c. Calculate Cov(X,Y).
I. Let {X n\ be a sequence of random variables wit h E(X,-? for n- 7n exists a C > 0 such that for n 1,2, 3,.. Show that X is cons istent for ?
Random variables z and y described by the PDF if x-+ yo 1 and x.> 0 and y, > 0 0 otherwise a Are x and y independent random variables? b Are they conditionally independent given max(x,y) S 0.5? c Determine the expected value of random variabler, defined byr xy.
Problem 7: Let X and Y be two jointly continuous random variables with joint PDF 4 (x y) otherwise a) Find P(0< Y< 1/2 I x-2) b) For what value of A is it true that P(0 < Y < ½ |X> A)-5/16
Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with the density function f(x)= λ exp (-Az), for >0. Consider T- and find its cumulative distribution function and density function.
2. If X and Y are independent random variables, X has a normal distribution with mean 2 variance 4, and Y has a chi-square distribution with 9 degrees of freedom, then find u such that P(X > 2+11,7)=0.01.
3. Suppose that X has pdf fx(x) = 3, x > 1 and Y has pdf 24» fy(y) = ¡2, x 〉 1. Suppose further that X and Y are inde- pendent. Calculate the P(X 〈 Y).
Suppose that the standard normal random variables X and Y are independent. Find P(0 < X<Y). 8 O 1 4T 0 1 8л Ala