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) (X¹₂) =(√²) X₁ X₂ Z₁ Z₂ = Z₁ (b) (e) (x₂) - (3/✓/² ¹/√²) (21) (3/√2 √2 X₁ (¹)(x)=(1) (2) (33) X₂ (1) (39)=(-1) (2) (22₁) = (1/² √²/₂) (²₂) (2) 1////2) Z₁ 2 2 √2 (3) (x₂) = (1 ² 0 Z₂ 1 Z3 (4) (Y) = (²¹) (²) 1

5. Suppose there are 2 populations, each with mean ₁ and 4₂ and standard deviation of ₁ and 2 respectively. The interest is to measure a random variable X by taking a random sample of size n from the first population. In a similar manner, another random variable Y is measured by independently taking a random sample of size m from the second population. Assuming n and m are sufficiently large, then demonstrate the following properties: a. X~N (μ₁, 1) b. 8~ N(H₂. /) c. (8-8)-№ (1₁ - 1₂ + 15t m

3.3.24. Let X1, X2 be two independent random variables having gamma distribu- tions with parameters a₁ = 3, B₁ = 3 and a2 = 5, B2 = 1, respectively. (a) Find the mgf of Y = 2X₁ +6X2. (b) What is the distribution of Y?

for the first part, could u tell me why the lower bound is 0 rather than x? We can see from the question that 0<x<y<1, if u set the lower bound=0, x might be bigger than y

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