Find Expected Value of X(n) from Uniform Distribution

[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

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

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

<br /> \int_0^{\theta} t f(t) \, dt<br />