If the difference between the expectation of the square of a random variable (E[X^2]) and the square of the expectation ...

Question

If the difference between the expectation of the square of a random variable (E[X^2]) and the square of the expectation of the random variable ((E[X])^2) is denoted by B, then:

A. B < 0
B. B >= 0
C. B > 0
D. B = 0

Accepted Answer

B. B >= 0 The quantity B = E[X^2] - (E[X])^2 is exactly the definition of the variance of the random variable X, written Var(X). Variance measures the spread of a distribution around its mean. Variance is always non-negative because it equals the expectation of the squared deviation (X - E[X])^2, and squares are never negative. Variance equals zero only when X is a constant (degenerate distribution), so B can equal 0.

If the difference between the expectation of the square of a random variable (E[X^2]) and the square of the expectation of the random variable ((E[X])^2) is denoted by B, then: A. B < 0 B. B >= 0 C. B > 0 D. B = 0