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I am working on a practice prelim question. It states:

Let $Y_1 < Y_2 < Y_3 < Y_4$ denote the order statistics of a random sample of size 4 from a distribution which is uniform on $(0,2)$. Find, with justification, the cumulative distribution function for $Y_3$, and use the cumulative distribution function to find the probability density function for $Y_3$

I know that once I get the CDF for $Y_3$ I can just take the derivative with respect to $Y_3$ to get the second part.

For finding the CDF would I need:

$$ P(\text{exactly 3 of the }X_i's \le x ) + P(\text{all of the }X_i's \le x )$$

I would then proceed to substitute the CDF (using the uniform distribution)?

I am not sure if this is the right idea.

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  • $\begingroup$ Dear @user0430 I see that, although you have already asked 18 question in this site and received answers in most of them, you have not mark a best answer in any of them. You can do it so by clicking on the checkmark next to the answer that you think is the one that helped you the most. Please read here for more detail. $\endgroup$ – Leo Sera Jul 2 '15 at 21:36
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We have $X_1,..,X_4~iid$ samples, therefore we can use the formula for the k-th order statistic $$\mathbb{P}(Y_{(k)}\le t) = \sum\limits_{m=k}^n \binom{n}{m}F(t)^m(1-F(t))^{n-m}$$ where F is the distribution function for $X_i\sim U(0,2)$ and n is the sample size. If you haven't seen this formula, yet, you can relatively easy proof it by looking at $S:=\sum\limits_{i=1}^n\textbf{1}_{X_i\le t}$, with $S\sim Bin(n,F(t))$.

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