Author Archives: Jonathan Mattingly

Let $ (X,Y)$ have joint density $f(x,y) = e^{-y}$, for $0<x<y$, and $f(x,y)=0$ elsewhere.

Are $X$ and $Y$ independent ?
Compute the marginal density of $Y$.
Show that $f_{X|Y}(x,y)=\frac1y $, for $0<x<y$.
Compute $E(X|Y=y)$
Use the previous result to find $E(X)$.

Three Random Variables

Let $X$, $Y$, and $Z$ be independent uniform $ (0,1)$.

Find the joint density of $XY$ and $Z^2$.
Show that $P(XY < Z^2) = \frac59$.

[Meester ex. 5.12.25]

A joint density example I

Let $ (X,Y) $ have joint density $f(x,y)=x e^{-x-y}$ when $x,y>0$ and $f(x,y)=0$ elsewhere. Are $X$ and $Y$ independent ?

[Meester ex 5.12.30]

Memory and the Exponential

Let $X$ have an exponential distribution with parameter $\lambda$. Show that

\[ P( X> t+ s \,|\, X>s) = P(X>t) \]

for all $s,t >0$. Explain why one might call this property of the exponential “the lack of memory”.

Hitting Zero

October 17, 2013

Consider a random walk making $2n$ steps, and let $T$ be the first
return to its starting point, that is
\[T =\ min\{1 ≤ k ≤ 2n : S_k = 0\},\]
and $T = 0$ if the walk does not return to zero in the first \(2n steps.

Show that for all $ 1 ≤ k ≤ n$ we have,

\[P(T = 2k) =\frac1{2k − 1} \begin{pmatrix} 2k\\k\end{pmatrix} 2^{-2k}\]

[Meetster Ex 3.3.2]

Geometric probability

October 17, 2013

in each case, consider a point picked uniformly randomly from the interior of the region. Find the probability density function for the $x$-coordinate.

The square with corner : $ (-2,0), (0,2), (2,0), (0,-2) $
The triangle with corners: $ (-2,0), (1,0), (0,2) $
The polygon with corners: $ (0,2),(2,1), (1,-1), (-1,0)$

[Pitman p277, # 12]

Expected value of Random Walk

October 17, 2013

Consider a random walk $S_i$, which begins at zero, and makes $2n$ steps.
Let $k < n$. Show that

$\mathbf{E}\Big( |S_{2k}| \ \Big|\ |S_{2k−1}| = r\Big) = r$;
$ \mathbf{E}\Big(|S_{2k+1}| \ \Big|\ |S_{2k}| = r\Big) = 1 \text{ if } r = 0, \text{ and } \mathbf{E}\Big(|S_{2k+1}| \ \Big|\ |S_{2k}| = r\Big) = r\text{ otherwise}.$

A p.m.f. and expectation example

September 18, 2013

Let $X$ be a random variable with probability mass function

\[p(n) = \frac{c}{n!}\quad \text{for $\mathbf{N}=0,1,2\cdots$}\]

and $p(x)=0$ otherwise.

Find $c$. Hint use the Taylor series expansion of $e^x$.
Compute the probability that $X$ is even.
Computer the expected value of $X$

[Meester ex 2.7.14]

Prime Dice

Suppose that we have a very special die which has exactly $k$ faces where $k$ is prime. The faces are numbered $1,\dots,k$. We throw the die once and see which number comes up.

What would be an appropriate outcome space and probability measure for this random experiment ?
Suppose that the events $A$ and $B$ are independent. Show that $\mathbf{P}(A)$ or $\mathbf{P}(B)$ is always either 0 or 1. Or in other wards $A$ or $B$ is always either the full space or the empty set.

[ from Meester, ex 1.7.32]

The chance a coin is fair

Suppose that I have two coins in my pocket. One ordinary, fair coin and one coin which has heads on both sides. I pick a random coin out of my pocket, throw it, and it comes up heads.

What is the probability that I have thrown the fair coin ?
If I throw the same coin again, and heads comes up again, what is the probability that I have thrown the fair coin ?
If instead of throwing the same coin again, I reach into my pocket and throw the second coin. If it comes up heads, what is the chance the first coin is the fair coin ?

[ Modified version of Meester, ex 1.7.35]

Independence of two hearts ?

Consider a deck of 52 cards. Let $A$ be the event that the first card is a heart. Let $B$ be the event that the 51st card is a heart.

What is $\mathbf{P}(A)$ ? What is $\mathbf{P}(B)$ ? Are $A$ and $B$ independent ?

Conditionally equally Likely

Let $A$ and $B$ be two events with positive probability. When does $\mathbf{P}(A|B)=\mathbf{P}(B|A)$ ?

Making a biased coin fair

Jack and Jill want to use a coin to decide who gets the remaining piece of cake. However, since the coin is Jack’s, Jill is suspicious that the coin is a trick coin which produced head with a probability $p$ which is not $\frac12$. Can you devise a way to use this coin to come to a fair decision as to who gets the cake ?

Selling the Farm

April 18, 2013

Two competing companies are trying to buy up all the farms in a certain area to build houses. In each year 10% of farmers sell to company 1, 20% sell to company 2, and 70% keep farming. Neither company ever sells any of the farms that they own. Eventually all of the farms will be sold. Assuming that there are a large number of farms initially, what fraction do you expect will be owned by company 1 ?

[Durrett “Elementary Probability”, p 159 # 39]

Computers on the Blink

April 18, 2013

A university computer room has 30 terminals. Each day there is a 3% chance that a given terminal will break and a 72% chance that that a given broken terminal will be repaired. Assuming that the fates of the various terminals are independent, in the long run what is the distribution of the number of terminals that are broken ?

[Durrett “Elementary Probability” p. 155 # 24]

Basic Markov Chains

April 16, 2013

In each of the graphs pictured, assume that each arrow leaving a vertex has an equal chance of being followed. Hence if there are thee arrows leaving a vertex then there is a 1/3 chance of each being followed.

For each of the six pictures, find the Markov transition matrix.
State if the Markov chain given by this matrix is irreducible and if the matrix is doubly stochastic.
If the Matrix is irreducible, state if it is aperiodic.
When possible (given what you know), state if each chain has a unique stationary distribution. If it is obvious that the system does not possess a unique stationary distribution, please state why.
For two of the chains it is easy to state what is this unique stationary distribution . Which two and what are the two stationary distributions?

conditional densities

March 30, 2013

Let $X$ and $Y$ have the following joint density:

\[ f(x,y)=\begin{cases}2x+2y -4xy & \text{for } 0 \leq x\leq 1 \ \text{and}\ 0 \leq y \leq 1\\ 0& \text{otherwise}\end{cases}\]

Find the marginal densities of $X$ and $Y$
find $f_{Y|X}( y \,|\, X=\frac14)$
find $ \mathbf{E}(Y \,|\, X=\frac14)$

[Pitman p426 # 2]

Expected max/min given min/max

March 30, 2013

Let $X_1$ and $X_2$ be the numbers on two independent fair-die rolls. Let $M$ be the maximum and $N$ the minimum of $X_1$ and $X_2$. Calculate:

$\mathbf{E}( M| N=x) $
$\mathbf{E}( N| M=x) $
$\mathbf{E}( M| N) $
$\mathbf{E}( N| M) $

Difference between max and min

March 19, 2013

Let $U_1,U_2,U_3,U_4,U_5$ be independent, each with uiform distribution on $(0,1)$. Let $R$ be the distance between the max and the min of the $U_i$’s. Find

$\mathbf{E} R$
the joint density of the max and the min of the $U_i$’s.
the $\mathbf{P}(R> .5)$

[pitman p355, #14]

A Joint density example II

March 19, 2013

If $X$ and $Y$ have joint density function

\[f(x,y)=\frac{1}{x^2y^2} \quad; \quad x \geq 1, y\geq 1\]

Compute the joint density fiction of $U=XY$, $V=X/Y$.
What are the marginal densities of $U$ and $V$ ?

[Ross p295, # 54]

Closest Point

Consider a Poisson random scatter of points in a plane with mean intensity $\lambda$ per unit area. Let $R$ be the distance from zero to the closest point of the scatter.

Find a formula for the c.d.f. and the density of $R$ and sketch their graphs.
Show that $\sqrt{2 \lambda \pi} R$ has the Rayleigh distribution.
Find the mean and mode of $R$.

[pitman p 389, # 21]

Joint Density of Arrival Times

Let $T_1 < T_2<\cdots$ be the arrival times in a Poisson arrival process with rate $\lambda$. What is the joint distribution of $(T_1,T_2,T_5)$ ?

Point of increase

Suppose $U_1,U_2, …$ are independent uniform $ (0,1) $ random variables. Let $N$ be the first point of increase. That is to say the first $n \geq 2$ such that $U_n > U_{n-1}$. Show that for $u \in (0,1)$:

\[\mathbf{P}(U_1 \leq u \ { and } \ N=n)= \frac{u^{n-1}}{(n-1)!}-\frac{u^{n}}{n!} \quad;\quad n \geq 2\]
$ \mathbf{E}(N)=e $

Some useful observations:

\[\mathbf{P}(U_1 \leq u \ { and } \ N=n) = \mathbf{P}(U_1 \leq u \ { and } \ N \geq n) -\mathbf{P}(U_1 \leq u \ { and } \ N \geq n+1)\]
The following events are equal
\[ \{U_1 \leq u \quad{ and } \quad N \geq n\} = \{U_{n-1}\leq U_{n-2} \leq \cdots \leq U_2\leq U_{1} \leq u \}\]
\[ \mathbf{P}\{U_2 \leq U_1 \leq u \}= \int_0^u \int_0^{u_1} du_2 du_1 \]

An example of min and change of variable