Question

Suppose that X1, ..., Xn form a random sample of size n from a continuous distribution with the following pdf: f(x) = 2x for 0 < x < 1
Let Y = max{X1, ...,Xn}. Evaluate E[Y]

Answer 1

Try using the equation function on the toolbar. I can't tell what you're asking.

Answer 2

You should approach this probability using the CDF.

\[ P(Y\le y)=P(\max\{X_1,\ldots,X_n\}\le y)\]

Now... What does \(\max\{X_1,\ldots,X_n\}\le y\) mean? Well, if the largest value \(X_i\) in the set is smaller than \(y\) , then any other value \(X_i\) MUST be also smaller than \(y\), because necessarily, all the other \(X_i\)'s that aren't the maximum will be smaller than the max (and hence of \(y\) as well).

So..
\[\begin{align} P(\max\{X_1,\ldots,X_n\}\le y) &= P(X_1\le y \cap X_2 \le y \cap \ldots \cap X_n \le y)\\
&= P(X_1\le y)P( X_2 \le y)\cdots P(X_n \le y) \\
&= \prod_{i=1}^n P(X_i\le y) \\
&=[F_{X_i}(y)]^n \end{align} \]
"Random sample" in mathematical statistics typically means I.I.D. So multiplying the probabilities from from the independence assumption, and in the step involving \(\prod\) is from assuming they're identically distributed.

To get the PDF now, just find the derivative
\[\begin{align} f_Y(y)&=\frac{d}{dy}[F_{X_i}(y)]^n\\
&= n[F_{X_i}(y)]^{n-1}\cdot f_{X_i}(y)\end{align} \]

So hopefully that helps to get you going on forward.