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Probability: An Introduction

Geoffrey Grimmett, Dominic Welsh

Chapter 3

Multivariate discrete distributions and independence - all with Video Answers

Educators

Chapter Questions

03:24

Problem 1

Let $X$ and $Y$ be independent discrete random variables, each having mass function given by
$$
\mathbb{P}(X=k)=\mathbb{P}(Y=k)=p q^{k} \quad \text { for } k=0,1,2 \ldots
$$
where $0<p=1-q<1$. Show that
$$
\mathbb{P}(X=k \mid X+Y=n)=\frac{1}{n+1} \quad \text { for } k=0,1,2 \ldots, n
$$${ }^{3} \mathrm{~A}$ similar fact is valid for an infinite sequence $A_{1}, A_{2}, \ldots$, namely that the mean number of events that occur is $\sum_{i=1}^{\infty} \mathbb{P}\left(A_{i}\right) .$ This is, however, harder to prove. See the footnote on p. 40 .

Shreya Kelly
Shreya Kelly
Numerade Educator
07:56

Problem 2

${ }^{3} \mathrm{~A}$ similar fact is valid for an infinite sequence $A_{1}, A_{2}, \ldots$, namely that the mean number of events that occur is $\sum_{i=1}^{\infty} \mathbb{P}\left(A_{i}\right) .$ This is, however, harder to prove. See the footnote on p. 40 .

Carlos Pinilla
Carlos Pinilla
Numerade Educator
07:10

Problem 2

Independent random variables $U$ and $V$ each take the values $-1$ or 1 only, and
$$
\mathbb{P}(U=1)=a, \quad \mathbb{P}(V=1)=b
$$
where $0<a, b<1$. A third random variable $W$ is defined by $W=U V$. Show that there are unique values of $a$ and $b$ such that $U, V$, and $W$ are pairwise independent. For these values of $a$ and $b$, are $U, V$, and $W$ independent? Justify your answer. (Oxford 1971F)

Mengchun Cai
Mengchun Cai
Numerade Educator
01:51

Problem 3

If $X$ and $Y$ are discrete random variables, each taking only two distinct values, prove that $X$ and $Y$ are independent if and only if $\mathbb{E}(X Y)=\mathbb{E}(X) \mathbb{E}(Y)$.

Aman Gupta
Aman Gupta
Numerade Educator
03:21

Problem 4

Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent discrete random variables, each having mass function
$$
\mathbb{P}\left(X_{i}=k\right)=\frac{1}{N} \quad \text { for } k=1,2, \ldots, N
$$
Find the mass functions of $U_{n}$ and $V_{n}$, given by
$$
U_{n}=\min \left\{X_{1}, X_{2}, \ldots, X_{n}\right\}, \quad V_{n}=\max \left\{X_{1}, X_{2}, \ldots, X_{n}\right\}
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
03:24

Problem 5

Let $X$ and $Y$ be independent discrete random variables, $X$ having the geometric distribution with parameter $p$ and $Y$ having the geometric distribution with parameter $r$. Show that $U=$ $\min \{X, Y\}$ has the geometric distribution with parameter $p+r-p r$.

Mengchun Cai
Mengchun Cai
Numerade Educator
03:06

Problem 6

Hugo's bowl of spaghetti contains $n$ strands. He selects two ends at random and joins them. He does this until no ends are left. What is the expected number of spaghetti hoops in his bowl?

Bryan Lynn
Bryan Lynn
Numerade Educator
02:36

Problem 7

Let $X_{1}, X_{2}, \ldots$ be discrete random variables, each having mean $\mu$, and let $N$ be a random variable which takes values in the non-negative integers and which is independent of the $X_{i}$. By conditioning on the value of $N$, show that
$$
\mathbb{E}\left(X_{1}+X_{2}+\cdots+X_{N}\right)=\mu \mathbb{E}(N)
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
05:46

Problem 8

Let $X_{1}, X_{2}, \ldots$ be independent, identically distributed random variables, and $S_{n}=X_{1}+X_{2}+$ $\cdots+X_{n} .$ Show that $\mathbb{E}\left(S_{m} / S_{n}\right)=m / n$ if $m \leq n$, and $\mathbb{E}\left(S_{m} / S_{n}\right)=1+(m-n) \mu \mathbb{E}\left(1 / S_{n}\right)$ if $m>n$, where $\mu=\mathbb{E}\left(X_{1}\right) .$ You may assume that all the expectations are finite.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:00

Problem 9

The random variables $U$ and $V$ each take the values $\pm 1$. Their joint distribution is given by
$$
\begin{gathered}
\mathbb{P}(U=+1)=\mathbb{P}(U=-1)=\frac{1}{2} \\
\mathbb{P}(V=+1 \mid U=1)=\frac{1}{3}=\mathbb{P}(V=-1 \mid U=-1) \\
\mathbb{P}(V=-1 \mid U=1)=\frac{2}{3}=\mathbb{P}(V=+1 \mid U=-1)
\end{gathered}
$$
(a) Find the probability that $x^{2}+U x+V=0$ has at least one real root.
(b) Find the expected value of the larger root, given that there is at least one real root.
(c) Find the probability that $x^{2}+(U+V) x+U+V=0$ has at least one real root.
(Oxford $1980 \mathrm{M})$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:45

Problem 10

A number $N$ of balls are thrown at random into $M$ boxes, with multiple occupancy permitted. Show that the expected number of empty boxes is ( $M-1)^{N} / M^{N-1}$

Aman Gupta
Aman Gupta
Numerade Educator
03:34

Problem 11

We are provided with a coin which comes up heads with probability $p$ at each toss. Let $v_{1}, v_{2}, \ldots, v_{n}$ be $n$ distinct points on a unit circle. We examine each unordered pair $v_{i}, v_{j}$ in turn and toss the coin; if it comes up heads, we join $v_{i}$ and $v_{j}$ by a straight line segment (called an edge), otherwise we do nothing. The resulting network is called a random graph.Prove that
(a) the expected number of edges in the random graph is $\frac{1}{2} n(n-1) p$,
(b) the expected number of triangles (triples of points each pair of which is joined by an edge) is $\frac{1}{6} n(n-1)(n-2) p^{3}$

Amany Waheeb
Amany Waheeb
Numerade Educator
01:24

Problem 12

Coupon-collecting problem. There are c different types of coupon, and each coupon obtained
is equally likely to be any one of the c types. Let Yi be the additional number of coupons
collected, after obtaining i distinct types, before a new type is collected. Show that Yi has the
geometric distribution with parameter (c ? i)/c, and deduce the mean number of coupons you
will need to collect before you have a complete set.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
01:24

Problem 13

In Problem 3.6.12 above, find the expected number of different types of coupon in the first n
coupons received.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
03:56

Problem 14

Each time you flip a certain coin, heads appears with probability p. Suppose that you flip the
coin a random number N of times, where N has the Poisson distribution with parameter ? and
is independent of the outcomes of the flips. Find the distributions of the numbers X and Y of
resulting heads and tails, respectively, and show that X and Y are independent.

Bryan Lynn
Bryan Lynn
Numerade Educator
16:48

Problem 15

Let $\left(Z_{n}: 1 \leq n<\infty\right)$ be a sequence of independent, identically distributed random variables with
$$
\mathbb{P}\left(Z_{n}=0\right)=q, \quad \mathbb{P}\left(Z_{n}=1\right)=p
$$
where $p+q=1$. Let $A_{i}$ be the event that $Z_{i}=0$ and $Z_{i-1}=1$. If $U_{n}$ is the number of times $A_{i}$ occurs for $2 \leq i \leq n$, prove that $\mathbb{E}\left(U_{n}\right)=(n-1) p q$, and find the variance of $U_{n}$. (Oxford $1977 \mathrm{~F}$ )

Mengchun Cai
Mengchun Cai
Numerade Educator
16:48

Problem 16

Let $\left(Z_{n}: 1 \leq n<\infty\right)$ be a sequence of independent, identically distributed random variables with
$$
\mathbb{P}\left(Z_{n}=0\right)=q, \quad \mathbb{P}\left(Z_{n}=1\right)=p
$$
where $p+q=1$. Let $A_{i}$ be the event that $Z_{i}=0$ and $Z_{i-1}=1$. If $U_{n}$ is the number of times $A_{i}$ occurs for $2 \leq i \leq n$, prove that $\mathbb{E}\left(U_{n}\right)=(n-1) p q$, and find the variance of $U_{n}$. (Oxford $1977 \mathrm{~F}$ )

Mengchun Cai
Mengchun Cai
Numerade Educator