How to find density function from this joint probability density function

Question

$$f(x,y) = \begin{cases} 2 & 0< x< y<1 \\ 0 &\text{otherwise} \end{cases} $$

I need to find

1. the density function of $y$ and
2. conditional pdf of $X$ given $Y$.

My answer is coming out to be

1. $2y$
2. $\frac1y$

Answer 1

The marginal density of $Y$ can be found by "averaging over" the $X$ values:

$$ f_Y(y) = \int_{0}^{y} 2 \, dx . $$

Once you have this marginal density you can combine it with the joint density to arrive at the conditional:

$$ f_{X|Y}(x, y) = \frac{f_{XY}(x, y)}{f_{Y}(y)} . $$

If you carry out these calculations you should get the answer you gave.

Answer 2

This is the kind of problem where it helps tremendously to have a visual image of the meaning of all those mathematical formulas in your book or notes.

• The graph of the pdf $f_X(u)$ of one random variable $X$ is a curve that lies (on or) above the $u$ axis, and the area under the curve is $1$. This is what all that gobbledygook $f_X(u) \geq 0$ and $\int_{-\infty}^\infty f_X(u)\, \mathrm du =1$. "But, but, ...., " you splutter, "the pdf of $X$ is $f_X(x)$, not $f_X(u)$ and so what you are saying makes no sense!" OK, sketch the graph of $f_X(x)$ on axes labeled $x$ and $f_X(x)$, and also the graph of $f_X(u)$ on axes labeled $u$ and $f_X(u)$. Apart from the labels on the axes, is there any difference between the two curves?
• Similarly, the graph of the joint pdf $f_{X,Y}(u,v)$ of two random variables $X$ and $Y$ is a surface that lies (on or) above the $u$-$v$ plane, and the volume of the solid bounded by the surface and the $u$-$v$ plane is $1$. This is what $f_{X,Y}(u,v) \geq 0$ and $\int_{-\infty}^\infty \int_{-\infty}^\infty f_{X,Y}(u,v)\, \mathrm du\, \mathrm dv =1$ means.
• For the given fixed real number $u_0$, the value of the marginal density $f_X$ of $X$ equals the area of the cross-section of the joint pdf solid when the solid is sliced by the plane $u = u_0$. Note that the cross-section of interest is the curve $f_{X,Y}(u_0,v)$ plotted as a function of $v$, and its area can be expressed as $\int_{-\infty}^\infty f_{X,Y}(u_0,v)\, \mathrm dv,$ that is, $$f_X(u_0) = \int_{-\infty}^\infty f_{X,Y}(u_0,v)\, \mathrm dv.\tag{0}$$ Similarly, for the given fixed real number $u_1$, the value of the marginal density $f_X$ of $X$ is $$f_X(u_1) = \int_{-\infty}^\infty f_{X,Y}(u_1,v)\, \mathrm dv\tag{1}$$ while for the given fixed real number $u_2$, the value of the marginal density $f_X$ of $X$ is $$f_X(u_2) = \int_{-\infty}^\infty f_{X,Y}(u_2,v)\, \mathrm dv\tag{2}$$ and so on. After a few more calculations of this kind, some smart-ass exclaimed "Hey Ma, I think I see a pattern here. For any fixed real number $u$, I can write $$f_X(u) = \int_{-\infty}^\infty f_{X,Y}(u,v)\, \mathrm dv\tag{*}$$ "Isn't that neat?" and the world has not been the same ever since. All the textbook writers began using $(*)$ without ever reminding their readers that all that you need to understand is that the value of the marginal pdf is the area of the cross-section of the joint pdf solid.
• For the given fixed real number $u_0$, the conditional pdf $f_{Y\mid X=u_0}(v\mid X=u_0)$ of $Y$ given that $X=u_0$ is shaped exactly like the the curve $f_{X.Y}(u_0,v)$ regarded as a function of $v$: the only difference is that the area under this curve might not be 1. But, any nonnegative function $g(v)$ whose area is finite (say, Area $A$) an be converted into a pdf by dividing $g(v)$ by $A$: $$g(v) \geq 0 ~\forall v~\text{and} \int_{-\infty}^\infty g(v)\, \mathrm dv = A \implies \int_{-\infty}^\infty \dfrac{g(v)}{A}\, \mathrm dv = 1\implies \dfrac{g(v)}{A} ~\text{is a pdf}.$$ Now, Eq. (0) tells us that the area under $f_{X,Y}(u_0,v)$ is just $f_X(u_0)$ and so the the conditional pdf $f_{Y\mid X=u_0}(v\mid X=u_0)$ of $Y$ given that $X=u_0$ is just $$f_{Y\mid X=u_0}(v\mid X=u_0) = \dfrac{f_{X,Y}(u_0,v)}{f_X(u_0)}.$$

  Put more simply, the conditional pdf is just the cross-sectional shape "normalized" to have area 1.

Applying these ideas to the question at hand, the joint pdf of $X$ and $Y$ is a surface at height $2$ above the plane, and the joint pdf solid is just a right prism of height $2$ whose base is a triangle with vertices at $(0,0), (0,1), (1,1)$. Its cross-section by the plane $v_0=0.2$ is a rectangle of height $2$ and base extending from $u=0$ to $u=0.2$. Consequently, the conditional pdf of $X$ given $Y=0.2$ is the uniform pdf $U(0,0.2)$, that is, $$f_{X\mid Y=0.2}(u\mid Y=0.2) = \begin{cases}5,&0 < u < 0.2,\\0, &\text{otherwise.}\end{cases}$$ I will leave it to the OP to have a Hey Ma! moment and come to the realization that for any fixed $y \in (0,1)$, we can write $$f_{X\mid Y=y}(u\mid Y=y) = \begin{cases}\frac 1y,&0 < u < y,\\0, &\text{otherwise.}\end{cases}$$ and not his glib answer $1/y$. Similarly, I will leave it to the OP to figure out why his glib calculation of $f_Y(y)=2y$ is incomplete, and how the area of cross-section idea makes finding the actual pdf $f_Y(y)$ an almost trivial calculation.

I now await the inevitable downvotes....