## Mathematics 340 - Fall 2015

### Homework

Exercises 7.8: 24, 31, 32(a), 35, 36

Also:

1. Suppose $X$ and $Y$ are jointly continuous random variables with joint probability density function $f_{X,Y}(x,y) = \begin{cases}8xy,& \text{if } 0 < x < y < 1,\\ 0,& \text{otherwise}.\end{cases}$ Find each of the following:
1. $E(X)$
2. $E(Y^2)$
3. $E(XY^3)$
2. Suppose $X$ and $Y$ are independent standard normal random variables. Find the distribution of $W = \sqrt{X^2 + Y^2}$.
3. Suppose $X$ and $Y$ are independent random variables with $Y$ uniformly distributed on $(0, 1)$ and $X$ uniformly distributed on $(0, 2)$. Find the probability density functions for each of the following:
1. $W = XY$
2. $W = \dfrac{Y}{X}$

For Problem Set due 11 November: 35 from Exercises 7.8 and 1 from the problems above

1. (a) $\frac{8}{15}$ (b) $\frac{2}{3}$ (c) $\frac{1}{3}$
2. For any $w > 0$, $P(W \le w) = \int\int_B \frac{1}{2\pi}e^{-\frac{x^2+y^2}{2}}dxdy$, where $B = \{(x, y) \ | \ x^2 + y^2 \le w^2\}$. Switching to polar coordinates, $P(W \le w) = \frac{1}{2\pi}\int_0^w \int_0^{2\pi} re^{-\frac{r^2}{2}}d\theta dr = 1 - e^{-\frac{w^2}{2}}$. Hence the probability density function of $W$ is $f_W(w) = \begin{cases}we^{-\frac{w^2}{2}},& \text{if } w > 0, \\ 0,& \text{otherwise}.\end{cases}$
1. $f_W(w) = \begin{cases}-\frac{1}{2}\log\left(\frac{w}{2}\right),& \text{if } 0 < w < 2, \\ 0,& \text{otherwise}.\end{cases}$
2. $f_W(w) = \begin{cases}1,& \text{if } 0 < w \le \frac{1}{2}, \\ \frac{1}{4w^2},& \text{if } w > \frac{1}{2}, \\ 0,& \text{otherwise}.\end{cases}$