Expected value of a max

Question

We have a roulette with the circumference $a$. We spin the roulette 10 times and we measure 10 distances, $x_1,\ldots,x_{10}$, from a predefined zero-point. We can assume that those distances are $U(0,a)$ distributed.

An estimation of the circumference $a$ is given:

$$a^* = \max(x_1,\ldots,x_{10})$$

To check whether it's biased or not I need to calculate:

$$E(a^*) = E(\max(x_1,\ldots,x_{10}))$$

How do I proceed? I don't know any rules for calculating the estimate of a $\max$.

Answer 1

Let $W=\max(X_1,\dots,X_{10})$. Then $W\le w$ if and only if $X_i\le w$ for all $i$. From this you can find the cdf of $W$, hence the density, hence the expectation.

Added: For any $i$, the probability that $X_i\le w$ is $\dfrac{w}{a}$.

So by independence, the cumulative distribution function $F_W(w)$ of $W$ is $\left(\dfrac{w}{a}\right)^{10}$ (for $0\le w\le a$)

It follows that the density function of $W$ is $\dfrac{1}{a^{10}}10w^9$ on $[0,a]$, and $0$ elsewhere.

Multiply this density function by $w$, integrate from $0$ to $a$ to find $E(W)$.

Answer 2

Hint: (1) Find $\mathbb P(a^*\lt t)$ for every $t$ in $(0,a)$. (2) Find a formula for $\mathbb E(a^*)$ as a function of the probabilities $\mathbb P(a^*\lt t)$.