Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If I have two dependent continuous random variables $X$ and $Y$ with known pdf's $f(x)$ and $f(y)$. How to calculate their join probability distribution $f(x, y)$?

For example if $Y = \sin{X}$ and I want to calculate the pdf of $Z$ where $Z = \frac{X}{Y}$ or $Z = X - Y$. So, how to find out $f(x, y)$ first?

share|cite|improve this question
up vote 2 down vote accepted

If $Y$ is a regular function of $X$, $(X,Y)$ cannot have a density since $(X,Y)$ is always on the graph of the function, which has measure zero. But you should not use this to compute the distribution of $Z$ a function of $X$.

Rather, you could use the fact that $Z$ has density $f_Z$ if and only if, for every measurable function $g$, $$ E(g(Z))=\int g(z)f_Z(z)\mathrm{d}z. $$ But, if $Z=h(X)$, $E(g(Z))=E(g(h(X)))$, and by definition of the distribution of $X$, $$ E(g(h(X)))=\int g(h(x))f_X(x)\mathrm{d}x, $$ where $f_X$ is the density of $X$. In the end, you know $h$ and $f_X$ and you must make the two expressions of $E(g(Z))$ coincide for every function $g$, hence a change of variable I will let you discover should yield an expression of $f_Z$ depending on $f_X$.

share|cite|improve this answer

You can't. You need to know how they are dependent.

share|cite|improve this answer
Nice, I know how they are dependent. I edited my question – Osama Gamal May 7 '11 at 1:55

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.