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- Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance. (a) Prove that X2 and Y are independent. Note that by symmetry, it will also follow that Y 2 and X are independent. (b) Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated (i.e. Cov(W, Z) > 0).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 X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z).Let 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 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)=1, x=0, zero elsewhere
- For any continuous random variables X, Y , Z and any constants a, b, show the following from the definition of the covariance: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 )Suppose X is a continuous random variable with p.d.f. fX(x) = kx2(1 − x) if 0 < x < 1. (b) Find the c.d.f FX(x) explicitly.
- An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table. y p(x, y) 0 5 10 15 x 0.01 0.06 0.02 0.10 5 0.04 0.16 0.20 0.10 10 0.01 0.15 0.14 0.01 (a) Compute the covariance for X and Y. (Round your answer to two decimal places.)Cov(X, Y) = (b) Compute ? for X and Y. (Round your answer to two decimal places.)? =Suppose X and Y are random variables with E[XY ] = 6, E[Y ] = 4 and E[X] = 5 Find Cov(X; Y )An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table. y p(x, y) 0 5 10 15 x 0 0.01 0.06 0.02 0.10 5 0.04 0.16 0.20 0.10 10 0.01 0.15 0.14 0.01 (a) Compute the covariance for X and Y. (Round your answer to two decimal places.)Cov(X, Y) = (b) Compute ρ for X and Y. (Round your answer to two decimal places.)ρ =
