Question

(squared mean error) The expected value of the random variable X is µ = 5 and the standard deviation σ = 2.
The aim is to predict the unknown value of the random variable with a constant and a prediction error
measured by the formula g (a) = E((X - a)²)

.
(a) Determine the value of constant a for which the prediction error is the lowest.
(b) Determine the minimum value of the prediction error.
(Hint: By applying the variance and expectation value calculation rules, g (a) can be converted to a
minimum is easier to figure out.)

Answer 1

g(a)= E((X-a)^2) = E(X^2 -2Xa -a^2)= E(X^2) - 2aE(X) -a^2 ------(1)

E(X) is the mean which is given as µ = 5.

(a) Prediction error is the lowest when the derivative of g(a) wrt a is minimum.

For this let us rewrite the above expression as E(X^2)- (E(X))^2 + (E(X))^2 -10a -a^2 ------(2)
Variance is given by = E(X^2)- (E(X))^2 = 4 (since SD=2 given)
So, E(X^2)- (E(X))^2 + (E(X))^2 -10a -a^2= 4+25-10a-a^2
dg(a)/da= -10-2a=0
=> a=-5

(b) Substitute a=-5 in equation 2 and find out the minimum value of prediction error