Chapter Questions
For $g$ from the above example show that$$\int_{0}^{1} \int_{0}^{1} g(x, y) \mathrm{d} x \mathrm{~d} y=-1, \quad \int_{0}^{1} \int_{0}^{1} g(x, y) \mathrm{d} y \mathrm{~d} x=1$$which shows that the iterated integrals may not be equal if Fubini's theorem condition is violated.
$$\text { Compute } \int_{[0,3] \times[-1,2]} x^{2} y \mathrm{~d} m_{2} \text {. }$$
$$\text { Compute the area of the region inside the ellipse } \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$
Take $A$ to be the square with corners at $(0,1),(1,0),(2,1),(1,2)$. Find the marginal densities of $f=\mathbf{1}_{A}$.
Let $f_{X, Y}(x, y)=\frac{1}{50}\left(x^{2}+y^{2}\right)$ if $0<x<2,1<y<4$ and zero otherwise. Find $P(X+Y>4), P(Y>X)$.
$$\text { Find } f_{X+Y} \text { if } f_{X, Y}=1_{[0,1] \times[0,1]}$$
Suppose that the joint density of $X, Y$ is $\mathbf{1}_{A}$ where $A$ is the square with corners at $(0,1),(1,0),(2,1),(1,2)$. Are $X, Y$ independent?
Find $P(Y>X)$ and $P(X+Y>1)$, if $X, Y$ are independent with $f_{X}=1_{[0,1]}, f_{Y}=\frac{1}{2} \mathbf{1}_{[0,2]}$
Let $f_{X, Y}=\mathbf{1}_{A}$, where $A$ is the triangle with corners at $(0,0),(2,0)$, $(0,1)$. Find the conditional density $h(y, x)$ and conditional expectation $\mathbb{E}(Y \mid X=1)$
Let $f_{X, Y}(x, y)=(x+y) \mathbf{1}_{A}$, where $A=[0,1] \times[0,1] .$ Find $\mathbb{E}(X \mid Y=y)$ for each $y \in \mathbb{R}$.