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. (-) (x) - (1) ² Z₁ Z₁ (b) (x₂) = (²/₂) (3/√2 1/√√2 (0) (X) = (¹/✓/³² ¹/✓³) (21) (31/1/2 0 (4¹) (x₂) = (²¹) 1) (²₂) (¹) () = (-1) (²) (2) (9) - (2) (1/ Y₁ (1/√√2 1/√2 = Y₂ Z₁ 2 2 (3) (X₂) = (² ² √²) (2/2 X2 1 1 0 Z3 2 (4) (Y) = (1 ¹) (²)
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. (-) (x) - (1) ² Z₁ Z₁ (b) (x₂) = (²/₂) (3/√2 1/√√2 (0) (X) = (¹/✓/³² ¹/✓³) (21) (31/1/2 0 (4¹) (x₂) = (²¹) 1) (²₂) (¹) () = (-1) (²) (2) (9) - (2) (1/ Y₁ (1/√√2 1/√2 = Y₂ Z₁ 2 2 (3) (X₂) = (² ² √²) (2/2 X2 1 1 0 Z3 2 (4) (Y) = (1 ¹) (²)
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Transcribed Image Text: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)
X₁
X2
=
(√2)
X₁
(b) (x₂)-(2₂2₂)
=
Z₁
X₁ 3/√√2
√2
X₂
=
IV2) (21)
0
1/√√2
1/√2 (Z₁
(c)
31
(¹) (X) - (11) (2)
(d
1
(1) (2)-()) (²)
=
Y₂
(2) (Y/₁2)
(12) = (1/✓/²2²
(3)
1/2) (²)
1/√√2 1/√√2
1/√√2 -1/√2,
2 2 √2
(X) - (² : ¹3) (2)
1
0
(4) (1/12)
(12) - (² 1) (²2)
=
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