Final answer:
The mean square error of the estimate of µy is En = Var(Y) + (Bias(μy))2 = 0.27 + (0.27 + (0.45)2 - μy)2
Step-by-step explanation:
The mean square error of the estimate of μy based on 45 independent samples of X can be calculated using the formula:
En = Var(Y) + (Bias(μy))2
Here, Var(Y) is the variance of Y, which is equal to the variance of X squared.
Since X is a continuous uniform random variable with parameters a = 0 and b = 0.9, the variance of X can be calculated using the formula:
Var(X) = (b - a)2/12
= (0.9 - 0)2/12
= 0.0675
Therefore, the variance of Y is equal to:
Var(Y) = Var(X2) = 4Var(X)
= 4 * 0.0675
= 0.27
Since X is a continuous uniform random variable, the expected value of X is equal to:
E(X) = (a + b)/2
= (0 + 0.9)/2
= 0.45
The bias of the estimate μy can be calculated as the difference between the expected value of Y and the true value of μy:
Bias(μy) = E(Y) - μy = E(X2) - μy
= Var(X) + (E(X))2 - μy
= 0.27 + (0.45)2 - μy
Therefore, the mean square error of the estimate of μy is:
En = Var(Y) + (Bias(μy))2
= 0.27 + (0.27 + (0.45)2 - μy)2