Exercises 7.8: 34

Also:

Find $\text{Corr}(X, Y)$ for each of the following joint probability density functions:

1. $f_{X,Y}(x, y) = \begin{cases} 8xy,& \text{if } 0 < y < x < 1,\\ 0,& \text{otherwise}.\end{cases}$.
2. $f_{X,Y}(x, y) = \begin{cases} x + y,& \text{if } 0 < y < 1, 0 < x < 1,\\ 0,& \text{otherwise}.\end{cases}$.
3. $f_{X,Y}(x, y) = \begin{cases} 24xy,& \text{if } x > 0, y > 0, x + y < 1,\\ 0,& \text{otherwise}.\end{cases}$.
4. $f_{X,Y}(x, y) = \begin{cases} e^{-y},& \text{if } 0 < x < y < \infty,\\ 0,& \text{otherwise}.\end{cases}$.

Answers:

1. 0.4924
2. $-\frac{1}{11}$
3. $-\frac{2}{3}$
4. $\frac{1}{\sqrt{2}}$