2. Consider two sets of multivariate random variables: x = and y = (y₁, y2, ....ya)', where E[x] combined random vector: Z= (*) ) = (x1, x2, y' Suppose z~ Npq(μ, Σ), where μl: = f1 = ... and Σ : - (x1, x2, ‚xp)' .... ₁ and E[y] =₂. Let z be the ‚Xp, Y1, Y2, ..., Yq)' a. Show that x ~ Ν,(μι, Σπ) and y b. Show that x and y are independent Σ11 12 Σ21 Σ22 Ng(μ2, Σ22) if and only if Σ12 = Σ21 = 0

Let Z₁ and Z₂ be independent standard normal random variables. Let pe [-1, 1]. Find a matrix L such that has a distribution. X = LZ N (1). (1))

Let (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 X^2 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).

-) Suppose X and Y are continuous random variables. The range of X is [1,3], the range of Y is [0, 1]. The joint pdf of X and Y be given by f(x, y) = 2xy³ - 2y³. Verify if X and Y independent random variables.