# Finding joint probability distribution of two dependent random variables?

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?

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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 $f_X$.