Unit 02 - Essential problems

Joint distributions

02

PMF calculations from a table

Suppose the joint PMF of $X$ and $Y$ has values given in this table:

$Y\downarrow X\to$	0	1	2	3
1	0.10	0.15	0	0.05
2	0.20	0.05	0.05	0.20
3	0.05	0	$x$	0.05

(a) Find $x$.

(b) Find the marginal PMF of $X$.

(c) Find the PMF of the random variable $Z=XY$.

(d) Find $P[X=Y]$ and $P[X>Y]$.

Link to original

Solution

Solutions---5190-02

05

Marginals and probability from joint PDF

Suppose $X$ and $Y$ have joint PDF given by:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}2e^{-(x+2y)}& \text{ if }x,y>0\\ 0& \text{ otherwise }\end{array}$$

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

(b) Find $P[X>Y]$.

Link to original

Solution

Solutions---5190-05

07

Grad student mentors

A university lab has an incoming group of researchers who need mentors. There are 4 graduate students, 2 undergraduate students, and the lab director who could serve as mentors. Suppose that exactly 3 of these 7 people will be selected as mentors.

(a) How many different groups of 3 people could be chosen to be the three mentors?

(b) Suppose exactly 2 graduate students and 1 undergraduate student are selected to be the three mentors. How many different groups of 3 people could be selected?

Let $X$ be the total number of graduate students chosen to be mentors, and $Y$ be the total number of undergraduate students chosen to be mentors.

(c) Construct the joint PMF of $X$ and $Y$: $P_{X,Y}(x,y)$

Link to original

Solution

Solutions---5190-07

12

Joint PMF with $y$-dependence

Suppose $X$ and $Y$ have the following joint PMF:

$$P_{X,Y}(x,y)=\{\begin{array}{cc}\frac{.05}{y}{(.95)}^{y-1}& x=\mathrm{1,2},\dots ,y;y=\mathrm{1,2},\dots \\ 0& \text{otherwise}\end{array}$$

(a) Find $P[Y<3]$.

(b) $P[X=1]=.1577$. Find $P[X=2]$.

Link to original

Solution

Solutions---5190-12

10

Soft-drink machine

A soft-drink machine has a random amount $Y$ in supply at the beginning of a given day and dispenses a random amount $X$ during the day (with measurements in gallons). It is not resupplied during the day, and therefore $X\le Y$. It has been observed that $X$ and $Y$ have a joint density given by:

$$f_{X,Y}(x,y)=\frac{1}{2},0\le x\le y\le 2$$

(a) Find the probability that the amount of soft-drink dispensed on a given day is greater than 1.5 gallons. (First compute the marginal PDF of $X$.)

(b) Find the probability that the amount of soda remaining in the machine at the end of the day is greater than $1/2$ gallon. That is, find $P[Y-X>1/2]$.

(c) Find the CDF of $W=Y-X$, the amount of soda remaining in the machine at the end of the day.

Link to original

Solution

Solutions---5190-10

11

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.

(a) Consider $W=\mathrm{Max}(X,Y)$, the time that will pass (beyond 12:00pm) before Alice and Bob will begin working together. Find the CDF of W.

(b) Find the probability Alice will have to wait more than 75 minutes for Bob.

Link to original

Solution

Solutions---5200-04

Independent random variables

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?

Link to original

Solution

Solutions---5200-01

Functions on two random variables

02

PDF of Min and Max

Suppose $X\sim \mathrm{Exp}(2)$ and $Y\sim \mathrm{Exp}(3)$ and these variables are independent. Find:

(a) The PDF of $W=\mathrm{Max}(X,Y)$

(b) The PDF of $W=\mathrm{Min}(X,Y)$

Link to original

Solution

Solutions---5210-02

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 $X+Y$.

Link to original

Solution

Solutions---5220-01

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.

Link to original

Solution

Solutions---5220-06

Covariance and correlation

01

Covariance and correlation

The joint PMF of $X$ and $Y$ is given by the table:

$Y\downarrow X\to$	0	1	2	3
1	$\frac{1}{15}$	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{1}{15}$
2	$\frac{1}{10}$	$\frac{1}{10}$	$\frac{1}{5}$	$\frac{1}{10}$
3	$\frac{1}{30}$	$\frac{1}{30}$	0	$\frac{1}{10}$

Compute:

(a) $E[X+Y]$ $$ (b) $E[{(X-Y)}^2]$ $$ (c) $\mathrm{Cov}[X,Y]$ $$ (d) $\rho [X,Y]$

Link to original

Solution

Solutions---5240-01

04

Correlation between overlapping coin flip sequences

Suppose a coin is flipped 30 times.

Let $X$ count the number of heads among the first 20 flips, and $Y$ count the heads in the last 20.

Find $\rho [X,Y]$.

Hint: Partition the flips into three groups of 10. Create three variables, counting heads, and express $X$ and $Y$ using these. What is the variance of a binomial distribution?

Link to original

Solution

Solutions---5240-04

07

Covariance etc. from joint density

Suppose $X$ and $Y$ are random variables with the following joint density:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}\frac{3}{2}(x^2+y^2)& x,y\in [\mathrm{0,1}]\\ 0& \text{ otherwise }\end{array}$$

Compute:

(a) $E[X]$ $$ (b) $E[Y]$ $$ (c) $E[XY]$ $$ (d) $\mathrm{Var}[X]$

(e) $\mathrm{Var}[Y]$ $$ (f) $\mathrm{Cov}[X,Y]$ $$ (g) $\rho [X,Y]$ $$ (h) Are $X$ and $Y$ independent?

(It is worth thinking through which of these can be computed in multiple ways.)

Link to original

Solution

Solutions---5240-07

Conditional distribution

01

Conditional density from joint density

Suppose that $X$ and $Y$ have joint probability density given by:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}{\displaystyle \frac{12}{5}x(2-x-y)}& x,y\in [\mathrm{0,1}]\\ & \\ 0& \text{ otherwise }\end{array}$$

(a) Compute $f_{X|Y}(x|y)$, for $y\in [\mathrm{0,1}]$.

(b) Compute $P[X>1/2|Y=y]$.

Link to original

Solution

Solutions---5250-01

02

From conditional to joint, and back again

Suppose we have the following data about random variables $X$ and $Y$:

$$\begin{array}{c}{f}_X(x)=\{\begin{array}{cc}3x^2& 0\le x\le 1\\ 0& \text{ otherwise }\end{array}\\ \\ f_{Y|X}(y|x)=\{\begin{array}{cc}2y/x^2& 0\le y\le x\\ 0& \text{ otherwise }\end{array}\end{array}$$

(a) Find the joint distribution $f_{X,Y}(x,y)$.

(b) Find $f_{X|Y}(x|y)$.

Link to original

Solution

Solutions---5250-02

03

Time till recharge

Let $X$ denote the amount of time (in hours) that a battery on a solar calculator will operate properly before needing to be recharged by exposure to light. The function below is the PDF of $X$.

$$f(x)=\{\begin{array}{cc}{\displaystyle \frac{50}{6}}{x}^{-3}& 2<x<10\\ & \\ 0& \text{ otherwise }\end{array}$$

Suppose that a calculator has already been in use for 5 hours. Find the probability it will operate properly for at least another 2 hours.

Link to original

Solution

Solutions---5250-03

Conditional expectation

02

Conditional distribution and expectation from joint PDF

Suppose that $X$ and $Y$ have the following joint PDF:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}cxy& 0<y<1,0<x<y\\ 0& \text{ otherwise }\end{array}$$

Notice that the range of possibilities for $x$ depends on the value of $y$.

First, show that $c=8$. Then compute:

(a) $f_X$ $$ (b) $f_{Y|X}$ $$ (c) $E[Y|X=0.5]$ $$ (d) $E[Y|X]$

Link to original

Solution

Solutions---5260-02