Suppose X has a discrete uniform distribution on {-n, -n+1, ..., -1, 0, 1, ..., n-1, n}. Then, EX for k = 0, 1, 2, ..., n is... (A) 2 (B) 0

Question

Suppose X has a discrete uniform distribution on {-n, -n+1, ..., -1, 0, 1, ..., n-1, n}. Then, E X² = k² for k = 0, 1, 2, ..., n is...

(A) 2
(B) 0

Answer 1

The conditional expectation E X² = k² is 0 for all values of k.

Step-by-step explanation:

The random variable X has a discrete uniform distribution on the set {-n, -n+1, ..., -1, 0, 1, ..., n-1, n}. We are asked to find the conditional expectation E X² = k² for k = 0, 1, 2, ..., n.

Since X can take both positive and negative values, we have to consider two cases separately:

Case 1: k = 0

When k = 0, X can only be 0. Hence, the conditional expectation E X² = 0² = 0.

Case 2: 0 < k ≤ n

When 0 < k ≤ n, there are two possible values of X that satisfy X² = k², i.e., X = k or X = -k. The conditional expectation EX is given by:

E X² = k² = (k + (-k))/2 = 0

Therefore, the answer is (B) 0.