Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $X$ be a random variable that is $X \sim \mathrm{Unif}(0,1) = 1$. Use a transformation method to find the pdf of $U = X(1 - X)$.

I tried solving for $X$ and I got $X = \dfrac{1 \pm \sqrt{1-4U}}{2}$

But the actual pdf is $\dfrac{2}{\sqrt{1-4U}}$, so I can't just take $X'$

share|improve this question

1 Answer 1

up vote 3 down vote accepted

Just do it systematically:

  • $U$ can take values in $(0,\frac14)$ and two different values of $X$ give the same $U$

  • $\Pr(X \le x)=x$ and $\Pr(X \ge x)=1-x$ for $x$ in $(0,1)$

  • $\Pr(U \le u) = \Pr\left(X \le \frac{1 - \sqrt{1-4u}}{2} \right) +\Pr\left(X \ge \frac{1 + \sqrt{1-4u}}{2} \right) = 1 - \sqrt{1-4u}$

  • $p(u) = \frac{d }{du} \Pr(U \le u)= \dfrac{2}{\sqrt{1-4u}}$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.