Given two independent random variable X and Y which both have uniform distribution over[0,1] I want to calculate PDF of $Z =\frac{X}{Y}$ and here is my solution: $\int_{-\infty}^{\infty}zf_X(yz)f_Y(y)dy = \int_{0}^{\frac{1}{z}}zdy = 1$ is this correct? one thing i am in doubt is the main relation that I have found over the internet: $\int_{-\infty}^{\infty}zf_X(yz)f_Y(y)dy$ is this correct?
Asked
Active
Viewed 53 times
0
-
You have to be more careful with limits of integration. For example, if $z<1$ then you won't be integrating up to $\frac 1 z$ because that goes beyond $(0,1)$. – Kavi Rama Murthy Apr 19 '19 at 09:05
-
@KaviRamaMurthy $0 \leq YZ \leq 1 \Rightarrow 0 \leq Y \leq \frac{1}{Z}$ is this wrong? – Peyman Tahghighi Apr 19 '19 at 09:14
-
But you cannot forget the extra condition $Y\leq 1$. – Kavi Rama Murthy Apr 19 '19 at 09:15
-
@KaviRamaMurthy so what should i do? – Peyman Tahghighi Apr 19 '19 at 09:20
-
If $z<1$ you have to integrate from $0$ to $1$, not $0$ to $1/z$. – Kavi Rama Murthy Apr 19 '19 at 09:23
-
Do these posts help you: https://math.stackexchange.com/questions/113295/pdf-of-a-quotient-of-uniform-random-variables, https://math.stackexchange.com/questions/2285127/using-cdf-to-calculate-ratio-of-two-uniformly-distributed-random-variables, https://stats.stackexchange.com/questions/15522/distribution-of-a-ratio-of-uniforms-what-is-wrong?noredirect=1&lq=1 ? – StubbornAtom Apr 19 '19 at 10:20