Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance

K. L. Chung, Farid AitSahlia

Chapter 7 Poisson And Normal Distributions - all with Video Answers

Educators

Chapter Questions

Problem 1

Suppose that a book of 300 pages contains 200 misprints. Use Poisson approximation to write down the probability that there is more than one misprint on a particular page.

Evelyn Cunningham

Numerade Educator

00:54

Problem 2

In a school where $4 \%$ of the children write with their left hands, what is the probability that there are no left-handed children in a class of $25 ?$

Sneha Ravi

Numerade Educator

01:45

Problem 3

Six dice are thrown 200 times by the players. Estimate the probability of obtaining isix different faces" $k$ times, where $k=0,1,2,3,4,5$.

Manik Pulyani

Numerade Educator

03:06

Problem 4

A home bakery made 100 loaves of raisin bread using 2000 raisins. Write down the probability that the loaf you bought contains 20 to 30 raisins.

Stanley Enemuo

Numerade Educator

01:19

It is estimated that on a certain island of 15 square miles there are 20 giant tortoises of one species and 30 of another species left. An
ecological survey team spotted 2 of them in an area of 1 square mile, but neglected to record which species. Use Poisson distribution to find the probabilities of the various possibilities.

Christopher Stanley

Numerade Educator

01:00

Problem 6

Find the maximum term or terms in the binomial distribution $B_{k}(n ; p)$, $0 \leq k \leq n .$ Show that the terms increase up to the maximum and then decrease. [Hint: take ratios of consecutive terms.

Amrita Bhasin

Numerade Educator

07:55

Problem 7

Find the maximum term or terms in the Poisson distribution $\pi_{k}(\alpha)$, $0 \leq k<\infty$. Show the same behavior of the terms as in No. 6 .

Abhirup Pal

Numerade Educator

01:32

Problem 8

Let $X$ be a random variable such that $P(X=c+k h)=\pi_{k}(\alpha)$, where $c$ is a real and $h$ is a positive number. Find the Laplace transform of $X$.

Sriram Soundarrajan

Numerade Educator

03:10

Problem 9

Find the convolution of two sequences given by Poisson distributions $\left\{\pi_{k}(\alpha)\right\}$ and $\left\{\pi_{k}(\beta)\right\} .$

Jacob Fry

Numerade Educator

View

Problem 10

If $X_{0}$ has the Poisson distribution $\pi(\alpha)$, then
$$
\lim _{\alpha \rightarrow \infty} P\left\{\frac{X_{\alpha}-\alpha}{\sqrt{\alpha}} \leq u\right\}=\Phi(u)
$$
for every $u$. [Hint: use the Laplace transform $E\left(e^{-\lambda\left(X_{\alpha}-\alpha\right) / \sqrt{\alpha}}\right)$, show that as $\alpha \rightarrow \infty$ it converges to $e^{\lambda^{2} / 2}$, and invoke the analogue of Theorem 9 of $\S$ 7.5.]

Shu Naito

Numerade Educator

01:33

Problem 11

Assume that the distance between cars going in one direction on a certain highway is exponentially distributed with mean value of 100 meters. What is the probability that in a stretch of 5 kilometers there are between 50 to 60 cars?

Dushyant Barot

Numerade Educator

02:36

Problem 12

On a certain highway the flow of traffic may be assumed to be Poissonian with intensity equal to 30 cars per minute. Write down the probability that it takes more than $N$ seconds for $n$ consecutive cars to pass by an observation post. [Hint: use (7.2.11).]

Linh Vu

Numerade Educator

02:56

Problem 13

A perfect die is rolled 100 times. Find the probability that the sum of all points obtained is between 330 and 380 .

Amany Waheeb

Numerade Educator

02:45

Problem 14

It is desired to find the probability $p$ that a certain thumbtack will fall on its flat head when tossed. How many trials are needed in order that we may be $95 \%$ sure that the observed relative frequency differs from $p$ by less than $p / 10 ?$ [Hint: try it a number of times to get a rough bound for $p .$.

Rakvi .

Numerade Educator

02:16

Problem 15

Two movie theaters compete for 1000 customers. Suppose that each customer chooses one of the two with "total indifference" and independently of other customers. How many seats should each theater have so that the probability of turning away any customer for lack of seats is less than $1 \%$ ?

Ivan Kochetkov

Numerade Educator

01:03

Problem 16

A sufficient number of voters are polled to determine the percentage in favor of a certain candidate. Assuming that an unknown proportion $p$ of the voters favor him and they act independently of one another,
how many should be polled to predict the value of $p$ within $4.5 \%$ with $95 \%$ confidence? [This is the so-called four percent margin of error in predicting elections, presumably because $<.045$ becomes $\leq .04$ by the rule of rounding decimals.

Victor Salazar

Numerade Educator

05:27

Problem 17

Write $\Phi((a, b))$ for $\Phi(b)-\Phi(a)$, where $a \leq b$ and $\Phi$ is the unit normal distribution. Show that $\Phi((0,2))>\Phi((1,3))$ and generalize to any two intervals of the same length. [Hint: $e^{-x^{2} / 2}$ decreases as $|x|$ increases.

Ahmad Reda

Numerade Educator

04:55

Problem 18

Complete the proof of (7.1.8) and then use the same method to prove (7.1.12). [Hint: $|\log (1-x)+x| \leq(1 / 2) \sum_{n=2}^{\infty}|x|^{n}=x^{2} /(2(1-|x|)$ )
hence if $|x| \leq 1 / 2$ this is bounded by $x^{2}$.]

Muhammad Saleem

Numerade Educator

00:54

Problem 19

Prove (7.1.13).

Monica Miller

Numerade Educator

06:22

Problem 20

Prove Chebyshev's inequality when $X$ has a density. [Hint: $\sigma^{2}(X)=$ $\left.\int_{-\infty}^{\infty}(x-m)^{2} f(x) d x \geq \int_{|x-m|>c}(x-m)^{2} f(x) d x .\right]$

Rashmi Sinha

Numerade Educator

05:46

Problem 21

Prove the following analogue of Chebyshev's inequality where the absolute first moment is used in place of the second moment:
$$
P(|X-m|>c) \leq \frac{1}{-} E(|X-m|)
$$

Mengchun Cai

Numerade Educator

07:46

Problem 22

Show that $\lim _{n \rightarrow \infty} P\left(\left|X_{n}\right|>\epsilon\right)=0$ for every $\epsilon$ if and only if given any $\epsilon$, there exists $n_{0}(\epsilon)$ such that
$$
P\left(\left|X_{n}\right|>\epsilon\right)<\epsilon \text { for } n>n_{0}(\epsilon)
$$
This is also equivalent to: given any $\delta$ and $\epsilon$, there exists $n_{0}(\delta, \epsilon)$ such that
$$
P\left(\left|X_{n}\right|>\epsilon\right)<\delta \text { for } n>n_{0}(\delta, \epsilon) .
$$
[Hint: consider $\epsilon^{\prime}=\delta \wedge \epsilon$ and apply the first form.

Shafiq Rehman

Numerade Educator

02:02

Problem 23

If $X$ has the distribution $\Phi$, show that $|X|$ has the distribution $\Psi$. where $\Psi=2 \Phi-1 ; \Psi$ is called the "positive normal distribution."

Victor Salazar

Numerade Educator

02:06

Problem 24

If $X$ has the distribution $\Phi$, find the density function of $X^{2}$ and the corresponding distribution. This is known as the "chi-square distribution" in statistics. [Hint: differentiate $\left.P\left(X^{2}<x\right)=2 / \sqrt{2 \pi} \int_{0}^{\sqrt{x}} e^{-u^{2} / 2} d u .\right]$

Ahmad Reda

Numerade Educator

01:33

Problem 25

Use No. 24 to show that
$$
\int_{0}^{\infty} x^{-1 / 2} e^{-x} d x=\sqrt{\pi}
$$
The integral is equal to $\Gamma(1 / 2)$, where $\Gamma$ is the gamma function defined by $\Gamma(\alpha)=\int_{0}^{\infty} x^{\alpha-1} e^{-x} d x$ for $\alpha>0 .$ [Hint: consider $E\left(X^{2}\right)$ in No. 24.]

Matthew Allcock

Numerade Educator

04:02

Problem 26

Let $\left\{\xi_{k}, 1 \leq k \leq n\right\}$ be $n$ random variables satisfying $0<\xi_{1} \leq \xi_{2} \leq$ $\cdots \leq \xi_{n} \leq t ;$ let $(0, t]=\cup_{k=1}^{l} I_{k}$ be an arbitrary partition of $(0, t]$ into subintervals $I_{k}=\left(x_{k-1}, x_{k}\right]$, where $x_{0}=0 ;$ and let $\tilde{N}\left(I_{k}\right)$ denote the
number of $\xi^{\prime}$ s belonging to $I_{k} .$ How can we express the event $\left\{\xi_{k} \leq\right.$ $\left.x_{k} ; 1 \leq k \leq l\right\}$ by means of $\tilde{N}\left(I_{k}\right), 1 \leq k \leq l ?$ Here, of course, $0<x_{1}<$
$x_{2}<\cdots<x_{n} \leq t .$ Now suppose that $x_{k}, 1 \leq k \leq l$, are arbitrary and answer the question again. [Hint: try $n=2$ and 3 to see what is going on; relabel the $x_{k}$ in the second part.]

Hunza Gilgit

Numerade Educator

00:51

Problem 27

Let $\{X(t), t \geq 0\}$ be a Poisson process with parameter $\alpha$. For a fixed $t>0$ define $\delta(t)$ to be the distance from $t$ to the last jump before $t$ if there is one, and to be $t$ otherwise. Define $\delta^{\prime}(t)$ to be the distance from $t$ to the next jump after $t$. Find the distributions of $\delta(t)$ and $\delta^{\prime}(t)$. [Hint: if $u<t, P\{\delta(t)>u\}=P\{N(t-u, t)=0\} ;$ for all $u>0$, $\left.P\left\{\delta^{\prime}(t)>u\right\}=P\{N(t, t+u)=0\} .\right]$

Victor Salazar

Numerade Educator

12:20

Problem 28

Let $\tau(t)=\delta(t)+\delta^{\prime}(t)$ as in No. 27 . This is the length of the betweenjump interval containing the given time $t$. For each $\omega$, this is one of the random variables $T_{k}$ described in $\S 7.2 .$ Does $\tau(t)$ have the same exponential distribution as all the $T_{k}$ 's? [This is a nice example where logic must take precedence over "intuition," and it is often referred to as a paradox. The answer should be easy from No. $27 .$ For further discussion at a level slightly more advanced than this book, see Chung, "The Poisson process as renewal process," Periodica Mathematica Hungarica, Vol. 2 (1972), pp. 41-48.

Robin Corrigan

Numerade Educator

01:56

Problem 29

Use Chebyshev's inequality to show that if $X$ and $Y$ are two arbitrary random variables satisfying $E\left\{(X-Y)^{2}\right\}=0$, then we have $P(X=Y)=1$, namely $X$ and $Y$ are almost surely identical. [Hint:
$P(|X-Y|>\epsilon)=0$ for any $\epsilon>0 .]$

Sagar Singh

Numerade Educator

01:44

Problem 30

Recall the coefficient of correlation $\rho(X, Y)$ from $\$ 6.3$. Show that if $\rho(X, Y)=1$, then the two "normalized" random variables
$$
\tilde{X}=\frac{X-E(X)}{\sigma(X)}, \quad \tilde{Y}=\frac{Y-E(Y)}{\sigma(Y)}
$$
are almost surely identical. What if $\rho(X, Y)=-1 ?$ [Hint: compute $E\left\{(\tilde{X}-\tilde{Y})^{2}\right\}$ and use No. 29.]

Ameer Said

Numerade Educator

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30