if the random variables x and y are independent, which of the following must be true? a) E[XY]>E[X]E[Y] b)cov(X,Y)<0 c) P(X=0|Y=0)=0 d) Cov(X,Y)=0
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if the random variables x and y are independent, which of the following must be true? a) E[XY]>E[X]E[Y] b)cov(X,Y)<0 c) P(X=0|Y=0)=0 d) Cov(X,Y)=0
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- Suppose X and Y are independent random variables with E(X) =2, E(Y)=3,V(X)=4,V(Y)=16. Finda)E(5X-Y) b)V(5X-Y) c)COV(3X+Y,Y) d)COV(X,5X-Y)Let X, Y be two Bernoulli random variables anddenote by p = P (X = 1), q = P (Y = 1) and r = P (X = 1, Y = 1). Prove that X and Y are independent if and only if r = pq.Suppose X and Y are random variables with E[XY ] = 6, E[Y ] = 4 and E[X] = 5 Find Cov(X; Y )
- 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 X1 and X2 be independent random variables for which P(Xi = 1) = 2/5 and P(Xi = 2) = 3/5 . Define U = X1 + X2 and V = X1 x X2. Calculate Cor(U, V )Let us say that we have jointly distributed random variables X and Y with E(X) = 3,E(Y) = 5, Var (X) = 1, Var(Y) = 3, and Cov (X, Y) = 2.Determine:a) E(2X + Y) b) Var(6X)
- Prove that for a continuous random variable X,E (aX+ b) = aE (X) + b.Assume Y=1+X+u, where X, Y, and u=v+X are random variables, v is independent of X; E(v)=0, Var(v)=1, E(X)=1, and Var(X)=2. a) E(u|X) b) E(Y|X)Q. For any random variables X and Y and the constant a, b, c and d show that Cov(aX + b, cY + d) = ac Cov(X, Y).
- Let X and Y be random variables. Suppose Var(X) = 1.6, Var(Y) = 1.2, and Cov(X, Y) = 0.6. Let Z = -1.2X + 1.9Y + 4.2. Calculate Var(Z).Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z).(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.