Question

Let X be a random variable that takes integer values and is symmetric, that is, P(X=k)=P(X=−k) for all integers k. What is the expected value of E[X]?

A. E[X]=0
B. E[X]=1
C. E[X]=k
D. E[X]=−k

Answer 1

The expected value of the symmetric random variable X is 0.

Step-by-step explanation:

The expected value of a random variable can be found by multiplying each possible value by its corresponding probability and summing them up. In this case, since X is symmetric, we have P(X = k) = P(X = -k).

Let's calculate the expected value:

1. E(X) = Σ xP(x)
2. Since P(X = k) = P(X = -k), we can combine the terms with positive and negative values.
3. E(X) = Σ (xP(x) + (-x)(P(-x))) = Σ (xP(x) - xP(-x)) = Σ x(P(x) - P(-x))
4. Since X takes integer values, we can write E(X) = Σ k(P(k) - P(-k))
5. Since P(X = k) + P(X = -k) = 1, we have P(k) = 1 - P(-k)
6. E(X) = Σ k[(1 - P(-k)) - P(-k)]
7. Since P(-k) + P(-k) = 1, we have P(-k) = 1/2
8. E(X) = Σ k[(1 - 1/2) - 1/2] = Σ k(1/2 - 1/2) = 0

Therefore, the expected value of X is 0.