Expected Maximum

[apalmer3]

I swear that I used to know this.

If you have an independent sample of size n, from the uniform distribution (interval [0,\theta]), how do you find the Expected Value of the largest observation(X(n))?

[statdad]

If X_{(n)} is the maximum in the sample, you first find its distribution. Since you have a random sample of size n , you can write


F(t) = \Pr(X_{(n)} \le t) = \prod_{i=1}^n \Pr(X_i \le t) = \left(\frac{t}{\theta}\right)^n


Differentiate this w.r.t. t to find the density f(t) , and the expected value is


\int_0^{\theta} t f(t) \, dt