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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 $\max$.

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    @Inquest Not really. Usually, one fers to order statistics when the joint distribution of the ordered sample is involed, not just the maximum of the sample.2012-12-13

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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 $E(W)$.

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    Not the integral of $wF(w)$, at least under the usual convention that $F_W(w)$ is the cumulative distribution function. The expectation is $\int_{-\infty}^{\infty}w f_W(w)\,dw$, where $f_W(w)$ is the density function of $W$. Hard to explain quickly. But recall that in the finite case, expectation is something like $\sum_{i=1}^n x_if(x_i)$, where $f(x_i)$ is the probability that the random variable takes on the value $x_i$. The integral is$a$generalized kind of "sum". If you are doing continuous random variables, I am sure the formula for their expectation is in your book.2012-12-14
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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 $\mathbb P(a^*\lt t)$.

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    You guessed incorrectly. Here $x$ is a running argument in $(0,a)$. Note that $a^*$ is not *estimated* and that *the biggest $x$* means nothing. I modified the notations.2012-12-13