I just had a question in my exam that I couldn't answer, I bugged.
You're given $X = \text{Median}(Y_1, Y_2, Y_3).$
$X$ follows an exponential distribution and you're asked to find the pdf of $X.$ My attempt was to find the cdf then derive.
Can someone explain me how I should have resolved it and what is the intuition behind the resolution.
Thanks you !
Comment continued (not Answer). From the standard exponential distribution (rate and mean both unity), I took a million samples of size $n = 3$ and found the median $h$ of each. Here is a histogram of the results. When you get your PDF you can compare it with the histogram. For a start, the distribution of $h$ is clearly not exponential. [The red curve is the PDF I got following the hints in my (real) Comment.]
Note: In case the formula for the PDF of the $i$th order statistic is not in your text or notes, look here at 7th slide or about halfway down the Wikipedia article on 'order statistics'.