Problems

01

Random point in a triangle

Consider a joint distribution that is uniform over the triangle with vertices $(\mathrm{0,0}),(\mathrm{0,1})$, and $(\mathrm{1,0})$. Suppose a point $(X,Y)$ is chosen at random according to this distribution.

(a) Find the joint PDF $f_{X,Y}$.

(b) Find the marginal PDFs for $X$ and $Y$.

(c) Are $X$ and $Y$ independent?

02

Factorizing the density

Consider two joint density functions for $X$ and $Y$:

$$\begin{array}{cc}{f}_1(x,y)& =6e^{-2x}{e}^{-3y}x,y>0,\\ f_2(x,y)& =24xyx,y\in [\mathrm{0,1}],x+y\in [\mathrm{0,1}].\end{array}$$

(Assume the densities are zero outside the given domain.)

Supposing $f_1$ is the joint density, are $X$ and $Y$ independent? Why or why not?

Supposing $f_2$ is the joint density, are $X$ and $Y$ independent? Why or why not?

03

Composite PDF from joint PDF

The joint density of random variables $X$ and $Y$ is given by:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}{e}^{-x-y}& x,y>0\\ 0& \text{otherwise}\end{array}$$

Compute the PDF of $Z=X/Y$.

04

Alice and Bob meeting at a cafe

Alice and Bob plan to meet at a cafe to do homework together. Alice arrives at the cafe at a random time (uniform) between 12:00pm and 1:00pm; Bob independently arrives at a random time (uniform) between 12:00pm and 2:00pm on the same day. Let $X$ be the time, past 12:00pm, that Alice arrives (in hours) and $Y$ be the time, past 12:00pm, that Bob arrives (in hours). So $X=0$ and $Y=0$ represent 12:00pm.

Find the joint PDF of $X$ and $Y$. (Hint: $X$ and $Y$ are independent.)

5

One car outlasts the other

Suppose that $X,Y\sim \mathrm{Exp}(0.1)$ are two independent exponential random variables.

(a) Find the joint PDF $f_{X,Y}$.

(b) Find the probability:

$$P[X-5>Y-15|X>5\text{AND}Y>15]$$