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Having X ~ Uniform(0,1), Y ~ Uniform(1,3) independent what's the pdf of Z = X/Y.

This means I can write the PDFs as follows $$f_X(x) = 1$$ for $ x \in \left(0,1\right)$ and 0 otherwise $$f_Y(y) = \frac{1}{2}$$ for $x \in \left(1,3\right)$ and 0 otherwise.

Using the following formula for division of independent random variables:

$$f_Z(u) = \int_0^{\infty}f_Y(y)f_X(yu)dy$$

$f_x(uy)$ is non-zero when $uy \in \left(0,1\right)$ so $y \in \left(0,\frac{1}{u}\right)$. The maximum $u$ is then $\frac{1}{3}$. What would be the right domain of integration in the formula above? Should the resulting PDF be then parameterized by $u$?

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  • $\begingroup$ Just to try out Mathematica's probability framework: D[Probability[ x/y <= z, {x [Distributed] UniformDistribution[{0, 1}], y [Distributed] UniformDistribution[{1, 3}]}], z] yields (for the pdf) 2 if z<1/3, (6-2z)/(4z)-(-z^2+6z-1)/(4z^2) if 1/3<z<1 and 0 elsewhere. $\endgroup$
    – PeterR
    Jan 23, 2013 at 15:46
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    $\begingroup$ @PeterR It would be more direct to try PDF[TransformedDistribution[x/y,{Distributed[x,UniformDistribution[]], Distributed[y, UniformDistribution[{1,3}]]}], z] $\endgroup$
    – Sasha
    Jan 23, 2013 at 16:04
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    $\begingroup$ Perhaps it is time to refer to Isaac Asimov's story The Feeling of Power again. $\endgroup$ Jan 23, 2013 at 16:08
  • $\begingroup$ But I'd like to know how to do this in hand. I don't get the integration of the Iverson bracket in the answer below. $\endgroup$
    – NumberFour
    Jan 23, 2013 at 17:35

1 Answer 1

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Actually, probability density functions for $X$ and $Y$ include indicator function (using Iverson bracket notation) $$ f_X(x) = [ 0 < x < 1] \qquad f_Y(y) = \frac{1}{2} [ 1 < y < 3] $$ Now: $$\begin{eqnarray} f_Z(z) &=& \int_0^\infty y f_Y(y) f_X(z \, y) \mathrm{d} y = \frac{1}{2} \int_1^3 y [ 0 < z \, y < 1 ] \mathrm{d} y \\ &=& [0<z<1] \int_1^3 \frac{y}{2} [z \, y < 1 ] \mathrm{d} y \\ &=& [0<z<1] \frac{1}{4} \left( \min\left(3^2,\frac{1}{z^2}\right) - 1 \right)\\ &=& [0<z<1] \min\left(2,\frac{1}{4}\left(\frac{1}{z^2}-1\right)\right) \end{eqnarray} $$

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  • $\begingroup$ I dont quite get the 4th step. How do I integrate the Iverson bracket? $\endgroup$
    – NumberFour
    Jan 23, 2013 at 16:47
  • $\begingroup$ Could you please explain? $\endgroup$
    – NumberFour
    Jan 23, 2013 at 17:21
  • $\begingroup$ @NumberFour For $0<z<1$ the Iverson bracket becomes zero for $y>\frac{1}{z}$. That is the effect of having the Iverson bracket in the integral is that the upper bound depends on $z$: $$ \int_1^3 \frac{y}{2} [ z y < 1] \mathrm{d} y = \int_1^{\min\left(3,z^{-1}\right)} \frac{y}{2} \mathrm{d} y$$ $\endgroup$
    – Sasha
    Jan 23, 2013 at 20:50
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    $\begingroup$ Oh, ok, could you please once again explain the 3rd step - how the Iverson bracket [ 0 < z < 1 ] came in front the integral? $\endgroup$
    – NumberFour
    Jan 23, 2013 at 20:52

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