Final answer:
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.