Exercise 2: Let X,Y and Z be random variables such that Cov(X, Y) < +oo. Show that Cov(X +E(X),Y + E(Y)) = E(Cov(X,Y||2) + Cov (E (X ||Z),E (Y||Z)).
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- Let X and Y be jointly distributed random variables. This exercise leads you through a proof of the fact that −1 ≤ ρX,Y ≤ 1. a) Express the quantity V(X − (σX/σY)Y) in terms of σX, σY, and Cov(X, Y). b) Use the fact that V(X − (σX/σY)Y) ≥ 0 and Cov(X, Y) = ρX,YσXσY to show that ρX,Y ≤ 1. c) Repeat parts (a) and (b) using V(X + (σX/σY)Y) to show that ρX,Y ≥ −1.Let X and Y be two continuous random variables having joint pdffX,Y (x, y) = (1 + XY)/4, −1 ≤x ≤1, −1 ≤y ≤1.Show that X ^2 and Y ^2 are independent.Prove that cov(X, Y) = cov(Y, X) for both discreteand continuous random variables X and Y.
- Prove that for a continuous random variable X,E (aX+ b) = aE (X) + b.Let X and Y be random variables, and a and b be constants. a) Prove that Cov(aX, bY) = ab Cov(X,Y). b) Prove that if a > 0 and b > 0, then ρaX,bY = ρX,Y. Conclude that the correlation coefficient is unaffected by changes in units.Let U1,U2 and U3 be mutually independent, uniform random variables on [0, 1]. LetX = min{U1,U2,U3}, Y = max{U1,U2,U3}.Find E[X | Y = y].
- X1 and X2 are independent random variables such that Xi has PDF fXi(x)={λiexp(−λix) when x≥0, 0 otherwise}. What is P[X2<X1]?Let X be a standard normal random variable, X ∼ N(0, 1). Find a real-valued scalar k such that: E[(aX^2 + b)^k ] = 1 where E[·] denotes expectation, and a, b are positive real numbers satisfying a + b < 1.Let X and Y be two jointly Gaussian real random variables, each with zero mean,variance 1, and correlation coefficient ρ ∈ (0, 1). Let a, b ∈ R be such that a2 + b2 = 1, and defineW := aX + bY .a. Find values for a and b to maximize the variance of W . Hint: Use eigendecomposition.b. Does the optimal W (from part a) have a probability density function? If yes, derive it. Ifnot, explain why.
- Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Cov (X,Y) = E[XY] - E[X] E[Y]Let X and Y be random variables, and a and b be constants. ???? a) Show that Cov [aX,bY] = abCov [X,Y] . b) Show that if a > 0 and b > 0, then the correlation coefficient between aX and bY is the same as the correlation coefficient between X and Y . c) Is the correlation coefficient between X and Y unaffected by changes in the units of X and Y ?The joint probability function of two discrete random variables X and Y is given by Ax,y) = c(2x+y), where x and y can assume all integers such that 0< x