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- 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.Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50Let X and Y be random variables. Suppose Var(X) = 1.6, Var(Y) = 1.5, and Cov(X, Y) = 0.1. Let Z = -1.2X + 1.2Y + 4.7. Calculate Var(Z).
- Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.Let X1, X2, X3,....., Xn be the independent identically distributed random variables and let Sm = X1 + X2 + .... + Xm. Assume that P(Sn = 0) = 0 and if m ≤ n then E(Sm/Sn) equals: Let X, Y be two Bernoulli random variables and denote 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.• Let {Xi, Yi}ni=1be a sample of n i.i.d. copies of (X, Y ). Based on this sample, we want to test whether X and Y are independent, i.e., whether r =pq
- Let pX(x) be the pmf of a random variable X. Find the cdf F(x) of X and sketch its graph along with that of pX(x) if pX(x)=x/15, x=1,2,3,4,5, zero elsewhereLet X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.Let 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).