Example 10-3. Show that if X', are the deviations of the random variables X and Y from their respective means, then 2 Σ X Y; 1 (i) r=1. 2N Ox Oy

3. Consider two random variables X₁ and X2 whose joint pdf is given by for x₁0, x2 > 0, x₁ + x2 < 2, { Find the pdf of Y = X₂ - X₁. f(x1, x₂) = NIT otherwise.

How were the correct answers of 1/45 and -2/73 attained?

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