Let X1 d= R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2.
(a) Find the state space of Y and derive the cdf FY and pdf fY
of Y .
(You may wish to use {X2 = i}, i = 0,1, as a partition and apply
the total probability formula.)
(b) Compute the mean and variance of Y in two different ways, one is through the pdf of Y and the other is using the sum of two independent random variables.
(c) Calculate E(U) and V(U).
(d) Find the state space of U and calculate the cdf FU of U.
(e) Calculate Cov(U, X2) and ρ(U, X2).




Let X1 d= R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y :=...
Let X1 d = R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.) (b) Compute the mean and variance of Y in two different ways, one is through the pdf of...
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Let X1 and X2 be independent n(0,1) random variables. Find the pdf of (X1 - X2)^2/2
5. Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Yı = X1 + X2 and ½ = X1 + ßX2. You are not given the constant β but it is known that Cov(Yi,Y) = 0. Find (a) the density of Y2 (b) Cov(Xy½),
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Let X and Y be two independent Bernoulli( 1/2 ) random variables. Define random variables U and V by U = X + Y and V = | (X - Y) | (abs. value)): (a) Find the joint probability mass function of (U, V ). Hints: note that U and V are taking integer values in {0, 1, 2} and {0, 1}, respectively. (b) Determine the covariance Cov(U, V ): (c) Find Var(U), Var(V ) and determine the correlation coeffcient p(U,...
Let X1~ exp(1) and X2 ~ exp(1) be independent and identically-distributed exponential random variables with rate 1. Let: Y = X1 + X2 , Z = X1 − X2 (a) What is the cdf of X1? (b) What is the joint pdf of (X1, X2)? (c) What is the joint pdf of (Y, Z)? (d) What is the marginal pdf of Z?
Given random variables X1, X2, Y with E[Y | X1, X2] =
5X1 + X1X2 and E[Y
2
| X1, X2] =
25X2
1X2
2 + 15, find
E[(X1Y + X2)
2
| X1, X2].
ㄨ竺Bin(2.1/4). Suppose X and Y are independent random variables. Find the expected value of YX. Hnt: Consider conditioning on the events (X-j)oj0,1,2. 9. Given random variables XI,X2, Y with E'Y | XiN;|-5X1 + X1X2 and Ep2 1 X1,X2] 25XX15, find 10. Let X and Y...