Determine if the random variables are statistically independent.
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- If X and Y have the joint probability distributionf(−1, 0) = 0, f(−1, 1) = 1 4 , f(0, 0) = 16 , f(0, 1) = 0, f(1, 0) = 112 , and f(1, 1) = 12 , show that (a) cov(X, Y) = 0;(b) the two random variables are not independent.Let U1, ....U5 be independent and standard uniform distibuted random variables given by P(U1 ≤ x) = x, 0 < x < 1 1. Compute the moment generating function E(e sU ) of the random variable U1. 2. Compute the moment generating function of the random variable Y = aU1 + U2 + U3 + U4 + U5 with a > 0 unknown. 3. Compute E(Y ) and V ar(Y ). 4. As an estimator for the unknow value θ = a we migth use as an estimator θb = 2 n Xn i=1 Yi − 4 = 2Y − 4. with Yi independent and identically distributed having the same cdf as the random variable Y discussed in part 2. Compute E(θb) and V ar(θb) and explain why this estimator is sometimes not very useful. 5.Give an upperbound on the probability P(| θb− a |> ) for every > 0.(Hint:Use Chebyshevs inequality!)Let X be a random variable with P(X = k) = 1/3 for k = −1, 0, 1. Let Y = X2.(a) What is the p.m.f. of Y?(b) Are X and Y independent?(c) Calculate E(XY ) − E(X)E(Y ).(d) True or false (for any random variables X and Y ): X and Y are independent if and onlyif E(XY ) − E(X)E(Y ) = 0 (or ρX,Y = 0).
- For a discrete random variable X if : Points ) P ( X = x ) = 0.1x , x = 2 , 3 , 5 , then E ( X2 ) =Let X be a random variable such that E(X2m) = (2m)!/(2mm!), m =1, 2, 3, . . . and E(X2m−1) = 0, m = 1, 2, 3, . . . . Find the mgf and the pdf of X.Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0
- For two random variables X and Y where X∼ exponential(0.5) and Y|X = x ∼ N(5,x2), find E(E(X|Y ))There are two traffic lights on a commuter's route to and from work. Let X1 be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning from work. Suppose that these two variables are independent, each with the pmf given in the accompanying table (so X1, X2 is a random sample of size n = 2). x1 0 1 2 p(x1) 0.1 0.2 0.7 ? = 1.6, ?2 = 0.44 (a) Determine the pmf of To = X1 + X2. to 0 1 2 3 4 p(to) (b) Calculate ?To. ?To = How does it relate to ?, the population mean? ?To = · ? (c) Calculate ?To2. ?To2 = How does it relate to ?2, the population variance? ?To2 = · ?2Let X and Y are independent Poisson random variables such that E(X) = E(Y)=2. Let Z=X+Y. Compute P(X=2|Z=3).