Expected value and variance of kth order statistic given maximum value

Question

Let X(1) < X(2) < X(3) < X(4) < X(5) be the order statistics corresponding to a random sample of size 5 from a uniform distribution on [0, θ], where θ ∈ (0, ∞). Prove that the variance of E[2X(3)| X(5)] <= variance of 2X(3)

Intuitively I understand that given a value, there is more information about X(3) and less variance. But how to mathematically prove it?

Answer 1

Hint: Since $2 X_{(3)}$ and $X_{(5)}$ are random variables in the same probability space , you may use the Law of Total Variance.