We measure the distance between continuous random variable X

We measure the distance between continuous random variable X and a constant b by (x-b)² . The closer b is to X, the smaller this quantity is. The quantity Q = E[ (X-b)² ] gives the expected distance between X and b. Find the value of b that minimizes Q. Use the second derivative test to verify that the value you found is a minimum.

Solution

Q = E(x-b)^2

Consider the function Q = (x-b)^2

dQ/db = 2(x-b)

d^2Q/db^2 = 2 >0

Hence when x = b the distance is minimum