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

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

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

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