Final answer:
The mean square error of the estimate of E(Y) based on 45 independent samples of X is 32.38.
Step-by-step explanation:
The random variable X is a continuous uniform (0, 9) random variable. The random variable Y is defined as Y = X^2. To find the mean square error of the estimate of E(Y) based on 45 independent samples of X, we need to find the variance of Y and divide it by the number of samples.
The variance of Y can be calculated as:
Var(Y) = Var(X^2) = E(X^4) - [E(X^2)]^2
Since X is a continuous uniform random variable between 0 and 9, the formulas for the theoretical mean and standard deviation of X^2 are:
E(X^2) = (9^3)/3 = 243
Var(X^2) = (9^5)/5 - (243)^2 = 1457.28
The mean square error of the estimate of E(Y) based on 45 independent samples of X is:
Mean Square Error = Var(Y)/n = 1457.28/45 = 32.38