Is $\max(X_1,X_2,X_3)$ an unbiased estimator

Question

Suppose that $X_1, X_2, X_3$ is a sample drawn from some population, if we use $\max(X_1, X_2,X_3)$ as an estimator of the population's mean, then can we say that its unbiased:

My approach was to let $X_{ i}$ be the max, $i=1,2,3$ and then deduce that $E(X_{ i})=\mu $, so did I miss anything?

Edit:

It appears due to the comments that $\max(X_1, X_2,X_3)$ follows a different distribution, given that the population is exponential how should I go around this sort of problem.

Thank you.

Answer 1

given that the population is exponential

just a detail...

As known, the distribution of the max is the product of the single CDF's, thus in your case, setting

$$Z=\max(X_1,X_2,X_3)$$

you get

$$F_Z(z)=\left(1-e^{-\theta z}\right)^3$$

derivating you get the density

$$f_Z(z)=3\theta e^{-\theta z}\left(1-e^{-\theta z}\right)^2$$

with expectation

$$\mathbb{E}[Z]=\int_0^{+\infty}3z\theta e^{-\theta z}\left(1-e^{-\theta z}\right)^2dz=\frac{11}{6\theta}$$

Concluding: the estimator is biased for the mean of the population, $\mu=\frac{1}{\theta}$, but with a correctable bias