let Z1, Z2, Z3 be iid ~N(0,1) .
Use moment generating function method to find Y=2Z1+3Z2+6Z3-2
let Z1, Z2, Z3 be iid ~N(0,1) . Use moment generating function method to find Y=2Z1+3Z2+6Z3-2
Problem 2. (Conditional Distribution of MVN) Let Z1, Z2, Z3 be i.i.d. N(0,1) dis- tributed random variables, and set X1 = 21 – Z3 X2 = 2Z1 + Z2 – 223 X3 = -221 +3Z3 1) What distribution does X = (X1, X2, X3)T follow? Specific the parameters. 2) Find out P(X2 > 0|X1 + X3 = 0).
Let Z ~ N(0,1) and let Y = Z2. Find the distribution of Y. Hint: Use moment generating function. Let X ~ N(j = 1, 02 = 4). If Y = 0.5*, find E(Y?). Hint: Use moment generating function.
4 points) Let Z1,Z2,...,Z1 be 11 independent N(O, 1) variables, and let Provide answers to the following to two decimal places Part a) Evaluate the moment generating function Mz2 (t) of Z2 at the point 0.23 Part b) Evaluate the moment generating function My(t) of Y at the point t = 0.31 . Part c) Find the mean of Y. Part d) Find the variance of Y.
Let U ⊂ C4 be the subspace U = {(z1, z2, z3, z4) ∈ C4: z1 + z2 + z3 + z4 = 0 and z1 = iz2}. (a) Find a basis for U. What is dim U? (b) Extend the basis from part (a) to a basis for C4. (c) Find a subspace W ⊂ C4 such that C4 = U ⊕ W. What is dim W?
Problem 1. (Bivariate Normal Distribution) Let Z1, Z2 be i.i.d. N(0,1) distributed random variables, and p be a constant between –1 and 1. define X1, X2 as: x3 = + VF5223X = v T14:21 - VF52 23 1) Show that, (X1, X2)T follows bivariate Normal distribution, find out the mean vector and the covariance matrix. 2) Write down the moment generating function, and show that when p= 0, X11X2.
Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y is the sum of independent random variables, compute both the mean and variance of Y. (b) Find the moment generating function of Y and use it to compute the mean and variance of Y.
Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y...
2. Let Z1, Z2, Zn be independent Normal(0,1) random variables (a) Find the MGF for Z for all i (b) Find the MGF for Σ_1 Z (c) If n is even, find the PDF for ΣΙ_1 z?
| Assume that Z1 and Z2 are two independent random variables that follow the standard normal dist ribution N(0,1), so that each of them has the density 1 (z) ooz< oo. e '2т X2 X2+Y2 Let X 212,Y 2Z1 2Z2, S X2Y2, and R (a) Please find the joint density of (Z1, Z2). (b) From (a), please find the joint density of (X,Y) (c) From (b), please find the marginal densit ies of X and Y. (d) From (b) and...
Let Z1, Z2,.., Zn be independent Normal(0,1) random variables (a) Find the MGF for Z for all i (b) Find the MGF for (c) If n is even, find the PDF for Σ
1. Let Z = (Z1, Z2, Z3) be a vector with i.i.d. N(0, 1) components. Let r be a constant with 0 < r < 1. Define X1 = √ rZ1 + √ 1 − rZ2 and X2 = √ rZ1 + √ 1 − rZ3. (a) Give the distribution of X1 and the distribution of X2. Find Cov(X1, X2). (b) Give the matrix A so that the vector X = (X1, X2) is a transform X = AZ. Give...