# order statistic (random sample from U(0,1) )

for a random sample $X_1 , X_2 .... X_5$ , from a Uniform (0,1) , isn't the distribution of median (say y) be given by $$f(y)= \frac{y^2 (1-y)^2}{\beta (3,3)}\,,\qquad 0 < y < 1$$ this can be obtained by directly using the result for the rth order statistic $$f(x)=\frac{F(x)^{r-1} (1-F(x))^{n-r}f(x)}{\beta (r,n-r+1)}$$ I need to prove that $p( y < 1/3 )= p (y >2/3)$,

I am not getting it true , is my result is correct ? Is there any other trick or property that can be used instead of integration ?

• Please add the self-study tag and read its tag wiki. Then please edit to show your attempt. Jan 8 '17 at 9:47
• What does $p(y < 1/3)$ mean? Isn't symmetry your friend (and nothing more?) Jan 8 '17 at 12:36

Your Beta(3,3) density being symmetric implies that, when $Y\sim \text{Beta}(3,3)$, then $(1-Y)\sim \text{Beta}(3,3)$. Thus $$\mathbb{P}(Y<1/3)=\mathbb{P}(1-Y<1/3)=\mathbb{P}(Y>2/3)$$