Sketching Region of Joint p.d.f. of X and Y -- Homework Help

Question

How can I sketch the joint p.d.f. of X and Y?

$f(x,y)=(\frac{1}{3}$ if $1<x<3$ and $0<y<1$ ; $\frac{1}{6}$ if $6<x<8$ and $0<y<1$ ; $0$ otherwise)

After this is graphed, find the marginal p.d.f.'s of X and Y?

Answer 1

For the PDF of the marginals, you just have to integrate:

$$ f_X(x) = \int f(x,y) dy= \begin{cases} \int_0^1 \frac 13 dy = \frac 13 &\text{ if }&1<x<3\\ \int_0^1 \frac 16 dy = \frac 16 &\text{ if }&6<x<8\\ 0&\text{ otherwise}\\ \end{cases} $$

$$ f_Y(y) = \int f(x,y) dx= \begin{cases} \int_1^3 \frac 13 dy + \int_6^8 \frac 16 dy = \frac23 + \frac 26 = 1 &\text{ if }&0<y<1\\ 0&\text{ otherwise}\\ \end{cases} $$

Bonus: you can check the relations $$ \int f_X(x)dx = \int_1^3 \frac 13 dx + \int_6^8 \frac 16 dx = 1 \\ \int f_Y(y)dy = \int_0^1 dy = 1 $$