If $X_1,\ldots,X_n\sim Uniform(0,10)$ and $Y=\max[X_1,\ldots,X_n]$, find $E(Y)$

Question

Given a set of n random variables $X_1,\ldots,X_n\sim Uniform(0,\alpha)$

Let $Y=\max[X_1,\ldots,X_n]$, What is the expectation $E(Y)$?

My question came when I saw an old question but I'm interested in the case where it is $\alpha=10$ for example so: $uniform(0,10)$, or more generally $uniform(0,\alpha)$ for any $\alpha$

What does it become in that case?

Edit: the questions is closed: "Please provide additional context, which ideally explains why the question is relevant to you and our community."

It is clearly mentioned in the first lines when I saw another answer by coincidence, I was curious to see how to generalize it. The part about "our community",well, if the mod who closed the question does not see it useful and can speak for the entire community, that's his own problem. Not mine to solve. Math SE was way nicer years ago. I hate the way this site is heading. It's trying to invite people to stop thinking, being curious or asking questions. good job mods!

Thanks

Answer 1

If $X_1, \ldots, X_n$ are i.i.d. $U(0,1)$, then $aX_1, \ldots, aX_n$ are i.i.d. $U(0,a)$ and $E(\max aX_i)=aE(\max X_i)$, so we have reduced the problem to that of the cited answer.

Answer 2

Given a finite collection of iid rv's $X_{1},...,X_{n}$ with cdf $F_{X}$. You have

You have for any random variable $Y$

$\displaystyle E(Y)=\int_{(0,\infty)}(1-F_{Y}(y))\,d\lambda(y) -\int_{(-\infty,0)}F_{Y}(y)\,d\lambda(y)$ . Where $\lambda$ is the Lebesgue Measure on $\Bbb{R}$. For simplicity you can ignore Lebesgue Integrability and treat everything as Riemann Integrals. See my answer here for a proof.

Now $F_{Y}(y)=\Bbb{P}(\max(X_{1},...,X_{n})\leq y)=\\\Bbb{P}\bigg((X_{1}\leq y)\cap (X_{2}\leq y)\cap...\cap(X_{n}\leq y)\bigg)=\bigg(\Bbb{P}(X_{1}\leq y)\bigg)^{n}=(F_{X}(y))^{n}$ .

Thus you can now calculate the expectation of $Y$ in terms of the cdf of $X_{1}$ alone if you substitute the value of $F_{Y}(y)$ in the formula for expectation I gave above.

In particular for $X_{i}\sim \text{unif}(0,a)$ the cdf $$F_{X}(y)=\begin{cases}0\,,y<0\\\frac{y}{a}\,,y\in (0,a)\\ 1\,,y\geq a\end{cases}$$ .

Thus after substituting you get

$$E(Y)=\int_{0}^{a}\bigg(1-\bigg(\frac{y}{a}\bigg)^{n}\bigg)\,dy$$