Problems

01

PDF of sum from joint PDF

Suppose the joint PDF of $X$ and $Y$ is given by:

$$f_{X,Y}=\{\begin{array}{cc}{\displaystyle \frac{8}{81}}xy& 0\le y\le x\le 3\\ 0& \text{otherwise}\end{array}$$

Find the PDF of $X+Y$.

02

$★$ Poisson plus Bernoulli

Suppose that:

• $X\sim \mathrm{Pois}(\lambda )$
• $Y\sim \mathrm{Ber}(p)$
• $X$ and $Y$ are independent

Find a formula for the PMF of $X+Y$.

Apply your formula with $\lambda =2$ and $p=0.3$ to find $P_{X+Y}(7)$.

Solution

Solutions---5220-02

03

PDF of sum of arbitrary uniforms

Suppose that:

• $X\sim \mathrm{Unif}[a,b]$
• $Y\sim \mathrm{Unif}[c,d]$
• $X$ and $Y$ are independent

Find the PDF of $X+Y$ in terms of the parameters $a,b,c,d$. You may assume that $b-a<d-c$.

04

$★$ Sums of normals

(a) Suppose $X,Y\sim 𝒩(\mu ,{\sigma }^2)$ are independent variables. Find the values of $\mu$ and $\sigma$ for which $X+X\sim X+Y$, or prove that none exist.

(b) Suppose $\mu =0$, $\sigma =1$ in part (a). Find $P[X>Y+2]$.

(c) Suppose $X\sim 𝒩(0,{\sigma }_X)$ and $Y\sim 𝒩(0,{\sigma }_Y)$. Find $P[X-3Y>0]$.

Solution

Solutions---5220-04

05

PDF of sum of uniforms

Let $X$ and $Y$ be independent copies of a $\mathrm{Unif}[\mathrm{0,1}]$ random variable. Let $W=X+Y$.

Find the PDF of $W$.

06

Lights on

An electronic device is designed to switch house lights on and off at random times after it has been activated. Assume the device is designed in such a way that it will be switched on and off exactly once in a 1-hour period. Let $X$ denote the time at which the lights are turned on and Y the time at which they are turned off. Assume the joint density for $(X,Y)$ is given by:

$$f_{X,Y}(x,y)=8xy,0<x<y<1$$

Let $W=Y-X$, the time the lights remain on during the hour.

(a) Find the range of $W$.

(b) Compute a formula for the CDF of $W$, i.e. $F_W(w)$.

(c) Find the probability the lights remain on for at least 40 minutes in some given hour.