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.
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 $S\sim Bin(n,F(t))$.