Suppose X is a random variable with PMF given by the following table: x = 1, 2, 3, 4

Suppose X is a random variable with E(X) = 0 and Var(X) = 2. What is E(X2 )? a). 1 b). 2 c). 3 d). 4 e). none of above

Consider a random variable X taking the values k1, k2, , km E R .. with probability ; Pn E [0, 1] 1. Write down the formula for the P1, P2, .. respectively, where p1 + P2 + expected value of f(X) for a given function f(-). + Pn ...

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)