I need the solution of this 8. Let 2 and Z, be independent standard normal random...
Let Z1 and 22 be independent standard normal random variables. Let pel-1, 1). Find a matrix L such that X = LZ has a 1 N 0 C) 1 distribution.
Exercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < l. Check that if X-Z and Y-p2+ VI-p-W then the pair (X, Y) has standard bivariate normal distribution with parameter ρ. Hint. You can use Fact 6.41 or arrange the calculation so that a change of variable in the inner integral of a double integral leads to the right density function.
#2 : Let X and Y be independent standard normal random variables, let Z have an arbitrary density function, and form Q = (X+ZY)/(V1+ Z2). Prove that Q also has a standard normal density function
xercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < 1 . Check that if X Z and Y-: ρΖ+ VI-P" W then the pair (X, Y) has standard bivariate normal distribution with parameter p. Hint. You can use Fact 6.41 or arrange the calculation so that a change of variable in the inner integral of a double integral leads to the right density function.
6. Let Z's be independent standard normal random variables. (a) Define X = Σ Z f X. (b) Define Y = 4 Σ zi. Find the mean and variance of Y. (Hint: Use the fact E(Z Z,)-0 for any i fj, i,j 1,2,3,4.) i. Find the mean and variance o i=1 4 i=1
4. Let X, Y, and Z be independent random variables, each with the standard normal distribution. Compute the following: (a) PIX Y> Z+2 (b) Var3X+4Y
4. Let X, Y, and Z be independent random variables, each with the standard normal distribution. Compute the following: (a) P[X + Y> Z +2 (b) Var3x 4Y;
3. Suppose X,,X2,, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X,X, X, UX , and V X- 9 9 Let X (x - x)2 i-1 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 32 VU Determine the distribution of the variable (c) (d) (e) Determine the distribution of the variable y (where Y is...
Let Y.Y2, ,Yn be independent standard normal random variables. That is, Y i-1,... ,n, are iid N(0, 1) random variables. 25 a) Find the distribution of Σ 1 Y2 b) Let Wn Y?. Does Wn converge in probability to some constant? If so, what is the value of the constant?
Assume that 2 and Z, are two independent random variables that follow the standard normal distribution N(0,1), so that each of them has the density - . - < < . Let X = 22 + 2 Z2, Y = 22 - Z2, S = x2 +Y, and R = xy (e) From (c), please find the densities of X? and Y? (f) From (d) and (e), please find the density of X2 +Y? (=S). (g) From (e), please find...