Q 6.1. Suppose Z = (Z₁, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., fori ≤ 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 onerandom variables, (1)–(4), on the right. Pair up according to "equal in distribution" and explainbriefly your reasoning.X₁(a)X₂(b) (x₂) - (2)(c) (x²) - (¹/1² 1/√²) (2₁)(3/√2=√2=(V2)Z₁(d) (x) - (1) (²)X2Y₁(9)-(-) (2)(1=1)Y₂(1)Y₁(²) (4) - (1/²(3)(1/√√21/√2-1/√2)(x₁) = (²₁ ² ✓/³²) (3)2 2 √21Z3Y₁→ ()-())(4)=Y₂1/2) (2)1)(2)

Given that be a standard multi-variate Gaussian random variable.For are i.i.d. random variables.

Answered: Q 6.1. Suppose Z = (Z1, Z2, Z3) is a…

A First Course in Probability (10th Edition)

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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

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