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Let $X,Y \sim$ Uniform$(0,1)$ be independent. Find the PDF for $X/Y$.

Let $Z=X/Y$. We want to find $F_z(z)=P(Z \leq z)=P(X/Y \leq z)$.

We can make $Y$ super small with fixed $X$, and conversely we can make $X$ really small with fixed $Y$. Thus it appears to me that we have $0<z<\infty$. I am struggling to find the subgraph of the unit square. We know that $X \leq Yz$. If $z = 1$, then we have a simple diagonal through the unit square. Increasing the value of $z$ shrinks what $Y$ can be since $X$ must be between $0$ and $1$. So we should see these lines fan out below the diagonal of the square as $z$ increases. Conversely, if $z$ approaches $0$, then we we limit the range of $X$, and so these lines fan out above the diagonal.

What I am having trouble with is putting this all together and calculating the integral itself.

StubbornAtom
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Lutterbach
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1 Answers1

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$$F_Z(z)=P[\frac{X}{Y} \leq z]=P[Y \geq \frac{X}{z}]$$

Drawing the line $Y=\frac{X}{z}$ in the unit square you see that CDF(z) is the area above this line thus

$$F_Z(z) = \begin{cases} 0, & \text{if $z<0$} \\ \frac{z}{2}, & \text{if $0\leq z<1$} \\ 1-\frac{1}{2z}, & \text{if $z \geq 1$} \end{cases}$$

To get PDF(z) just derivate F

Not any integral is needed. All the calculations can be done evaluating triangles' areas.

tommik
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