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 ?

  • 1
    $\begingroup$ Please add the self-study tag and read its tag wiki. Then please edit to show your attempt. $\endgroup$
    – Glen_b
    Jan 8, 2017 at 9:47
  • 2
    $\begingroup$ What does $p(y < 1/3)$ mean? Isn't symmetry your friend (and nothing more?) $\endgroup$
    – wolfies
    Jan 8, 2017 at 12:36

1 Answer 1


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 $$\mathbb{P}(Y<1/3)=\mathbb{P}(1-Y<1/3)=\mathbb{P}(Y>2/3)$$

  • $\begingroup$ Good but this is self study so we should just hint at first. wolfies hint suffices until the OP comes back with some other questions. $\endgroup$ Jan 8, 2017 at 21:09
  • $\begingroup$ I know that is what bothers me about self study questions. I have a recent discussion question about it (the self study tag) on CV meta. My current thinking is to wait for the OP and after that if he is still having problems I give an answer. If days go by with no response from the OP I might then give the answer also. This trick of yours is worth knowing. $\endgroup$ Jan 8, 2017 at 21:28
  • $\begingroup$ @Xian thank you so much xian ,.,., and yes i agree with u that a lesser hint would still have been confusing one , so thanks for the answer $\endgroup$
    Jan 9, 2017 at 4:35

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