Problem 7. Let X₁,..., Xn be independent N(0,0²) random variables for some unknown real 0. Find a Bayes estimator of for a N(μ, 7² prior. Here σ, µ and 7 are known.

Complete the following multiple choice questions. No work will be graded in the bonus problem. (a) E(X²)= 16. Compute E[-X(3X – 2)]. Assume that X is a random variable with E(X)= -3 and | (i) -54 (ii) -48 (iii) -33 (iv) -27

Q 6.1. Suppose Z = (Z₁, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i ≤ 3, Zi~ N(0, 1) are i.i.d. random variables. Each of the random variables, (a)–(d), on the left is equal in distribution to exactly one random variables, (1)–(4), on the right. Pair up according to "equal in distribution" and explain briefly your reasoning. X₁ (a) X₂ (b) (x₂) - (2) (c) (x²) - (¹/1² 1/√²) (2₁) (3/√2 = √2 = (V2) Z₁ (d) (x) - (1) (²) X2 Y₁ (9)-(-) (2) (1=1) Y₂ (1) Y₁ (²) (4) - (1/² (3) (1/√√2 1/√2-1/√2) (x₁) = (²₁ ² ✓/³²) (3) 2 2 √2 1 Z3 Y₁ → ()-()) (4) = Y₂ 1/2) (2) 1) (2)

2. Suppose one has n non-degenerate random variables X1, X2,, X, so that X1+ X2 + · · · + Xn = L for some constant L. (Recall that a random variable is non-degenerate if it is not a constant in disguise.) Show that there must be at least one pair of indices i j so that p(X;, X;) < 0.

Q 6.1. Suppose Z = (Z1, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i ≤ 3, Zi~ N(0, 1) are i.i.d. random variables. Each of the random variables, (a)-(d), on the left is equal in distribution to exactly one random variables, (1)-(4), on the right. Pair up according to "equal in distribution" and explain briefly your reasoning. (a) (X₂) (¹) (X) = (2) 1/√2 (Ⓒ) (x₂) - (3/1² ¹1/1²) (2) (c = 2 0 = 2 Z₁ 1) (²) 3 (4¹) (X) = (²¹) (1) (2) (3) Y₁ 1 (29) - ()) (2) = Y₂ 1 Y₁ (121) = (1/² √₂) (2) (1/√2 1/√2 /2 −1/√√2, Y₂ X₁ X₂ = 22 1 1 2 0 Y₁ 2 1 (4) (2)) = (²) (²) 1 1 Z₁ Z₂ Z3