Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\max$.

share|improve this question
    
Order Statistics? –  Inquest Dec 13 '12 at 16:32
    
@Inquest ?? $ $ –  Did Dec 13 '12 at 16:37
    
@did, isn't what Andre Nicolas suggested called Order Statistics? –  Inquest Dec 13 '12 at 16:38
    
@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. –  Did Dec 13 '12 at 18:46
add comment

2 Answers 2

up vote 1 down vote accepted

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 $E(W)$.

share|improve this answer
    
The detour by the densities is not needed. –  Did Dec 13 '12 at 16:36
    
Certainly true. It is a matter of guessing what is likely the more comfortable path for the student in a typical introductory course. –  André Nicolas Dec 13 '12 at 16:39
    
Sorry but I don't understand your answer. What is $w$? Where did you get it from? Is "iff" a typo or some special notation? –  Dimme Dec 13 '12 at 16:46
    
$w$ is a variable. Random variable $W$ is the maximum. I am using $n$ instead of $10$. Have expanded the "iff". The cdf of $W$, by hint given above, is $(w/a)^n=(1/a^n)w^n$ (on $[0,a]$). Density is therefore $(1/a^n)nw^{n-1}$. For expectation, multiply by $w$, integrate from $0$ to $a$. Get $\dfrac{na}{n+1}$, in your case $\dfrac{10a}{11}$. So estimator not unbiased, can make it unbiased by multiplying by $11/10$. –  André Nicolas Dec 13 '12 at 17:17
    
I have done some more exercises now and I understand more. You answer was actually quite helpful. I have one more question if you don't mind: Why is E(W) the integral of f(w) * w? I don't seem to find a formula that looks like that in my book. –  Dimme Dec 14 '12 at 0:50
show 1 more comment

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 $\mathbb P(a^*\lt t)$.

share|improve this answer
    
I guess that $P(a^* < x) = 0$ for any $x$ since $a^*$ is estimated to be the biggest $x$ out there and no other $x$ can be bigger that the biggest. So far so good, but I don't understand your second hint. How can an expected value be a function of probabilities 0? –  Dimme Dec 13 '12 at 16:42
    
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. –  Did Dec 13 '12 at 18:48
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.