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$$X_1, X_2, \dots, X_n\sim Unif(0, \theta), iid$$

and suppose

$$\hat\theta = \max\{X_1, X_2, \dots, X_n\}$$

How would I find the probability density of $\hat\theta$?

Thank you!

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are the Xs Independent? – Inquest Feb 24 '13 at 22:12
Here is a question for you: did you ask all your questions on MSE with no indication whatsoever on what you understood of the problem or what you tried before asking? – Did Feb 24 '13 at 22:19
@Inquest, yes they are! Sorry that I forgot to point that out. – Enzo Feb 24 '13 at 23:06
@Did, I am not sure what you are referring to. I am sorry if I left out what I have tried in my question. I knew that I was supposed to find the CDF of $\hat\theta$ and then differentiate it, but I got confused at the step of calculating $P\{max\{X_1, X_2, \dots, X_n\} < x\}$. – Enzo Feb 24 '13 at 23:08
The question is crystal clear. That you evade it is telling. I suggest to stop being sorry and to start putting some personal input in your questions. – Did Feb 25 '13 at 7:23
up vote 8 down vote accepted

\begin{align} P(Y\leq x)&=P(\max(X_1,X_2 ,\cdots,X_n)\leq x)\\&=P(X_1\leq n,X_2\leq n,\cdots,X_n\leq x)\\ &\stackrel{ind}{=} \prod_{i=1}^nP(X_i\leq x )\\&=\prod_{i=1}^n\dfrac{x}{\theta}\\&=\left(\dfrac{x}{\theta}\right)^n \end{align}

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In line 2, are the ns supposed to be xs or is this correct as is? – kram1032 Jul 26 '14 at 20:50

Let random variable $W$ denote the maximum of the $X_i$. We will assume that the $X_i$ are independent, else we can say very little about the distribution of $W$.

Note that the maximum of the $X_i$ is $\le w$ if and only if all the $X_i$ are $\le w$. For $w$ in the interval $[0,\theta]$, the probability that $X_i\le w$ is $\frac{w}{\theta}$. It follows by independence that the probability that $W\le w$ is $\left(\frac{w}{\theta}\right)^n$.

Thus, in our interval, the cumulative distribution function $F_W(w)$ of $W$ is given by $$F_W(w)= \left(\frac{w}{\theta}\right)^n.$$ Differentiate to get the density function of $W$.

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There appears to be a TeX error. I can't edit it because it's just a single backslash that needs removal but math.SE asks me to change the post by at least 6 symbols. – kram1032 Jul 26 '14 at 20:51
Thank you for telling me, yes I had written \w for w. – André Nicolas Jul 26 '14 at 21:02

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