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

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Duplicate of this ? – Sasha Feb 25 '12 at 18:37
up vote 3 down vote accepted

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.

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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

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