standard deviation for a sample from a uniform distribution?

Question

Let $X_1,X_2,...,X_n$ be a random sample from $U(0,1)$. I need to find the standard deviation of $max${$X_1,X_2,...,X_n$}. I am not sure how to begin with this problem?

Answer 1

You begin the problem by identifying the distribution's CDF.

$$\mathsf P(\max\limits_{i=1}^n \{X_i\} \leq x) = \mathsf P(\bigcap_{i=1}^n (X_i\leq x))$$

Then you use the fact that for any surely non-negative continuous random variable $$\mathsf E(Y)=\int_0^\infty \mathsf P(Y>y)\operatorname d y$$

To find $\mathsf{Var}(\max \{X_i\}) = \mathsf E\Big(\big(\max \{X_i\}\big)^2\Big)-\Big(\mathsf E\big(\max \{X_i\}\big)\Big)^2$