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.

Suppose $x_1, x_2$ are IDD random variables uniformly distributed on the interval $(0,1)$. What is the pdf of the quotient $x_2 / x_1$?

share|improve this question
Duplicate of this ? –  Sasha Feb 25 '12 at 18:37

1 Answer 1

Because $X_2$ is positive with almost surely, cumulative distribution function for $Z=X_1/X_2$ is $$ F_Z(z) = \mathbb{P}(Z \leqslant z) = \mathbb{P}(X_1 \leqslant z X_2) = \mathbb{E}_{X_2}\left( \mathbb{P}(X_1 \leqslant z X_2 | X_2)\right) = \mathbb{E}_{X_2}\left( F_{X_1}(z X_2)\right) $$ Clearly $F_Z(z)=0$ for $z\leqslant 0$, so assume $z > 0$ $$ F_Z(z) = \int_0^1 \left\{ \begin{array}{cl} z x_2 & 0 < x_2 <1/z \\ 1 & x_2 > 1/z \end{array} \right. \mathrm{d} x_2 = \frac{z}{2} \left(\frac{1}{\max(z,1)}\right)^2 + \left(1 - \frac{1}{\max(z,1)}\right) = \left\{\begin{array}{cl} \frac{z}{2} & 0< z\leqslant 1 \\ 1-\frac{1}{2z} & z > 1 \end{array} \right. $$ The probability density is obtained by differentiation.

share|improve this answer
You could also draw what is happening in the [0,1]x[0,1] square and identify z/2 and 1/(2z) as the areas of some simple triangles. –  Did Feb 25 '12 at 19:01

Your Answer


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.