Limit Theorems

Suppose that X is a random variable with mean and variance both equal to 20. What can be said about P{0 < X < 40}? 

Let X₁, ... , X₂₀ be independent Poisson random variables with mean 1. 

(a) Use the Markov inequality to obtain a bound on 

$P\left\{\underset{1}{\overset{20}{\sum X_i>15}}\right\}$

(b) Use the central limit theorem to approximate 

$P\left\{\underset{1}{\overset{20}{\sum X_i>15}}\right\}$                                                                                 

Fifty numbers are rounded off to the nearest integer and then summed. If the individual round-off errors are uniformly distributed over (−.5, .5), approximate the probability that the resultant sum differs from the exact sum by more than 3.

A die is continually rolled until the total sum of all rolls exceeds 300. Approximate the probability that at least 80 rolls are necessary.

From past experience, a professor knows that the test score taking her final examination is a random variable with a mean of$75$.

$\left(a\right)$Give an upper bound for the probability that a student’s test score will exceed$85$.

$\left(b\right)$Suppose, in addition, that the professor knows that the variance of a student’s test score is equal$25$. What can be said about the probability that a student will score between $65$and$85$?

$\left(c\right)$How many students would have to take the examination to ensure a probability of at least $.9$that the class average would be within $5$of$75$? Do not use the central limit theorem.

Use the central limit theorem to solve part $\left(c\right)$of the problem$8.2$.

A person has 100 light bulbs whose lifetimes are independent exponentials with mean 5 hours. If the bulbs are used one at a time, with a failed bulb being replaced immediately by a new one, approximate the probability that there is still a working bulb after 525 hours. 

A.J. has $20$jobs that she must do in sequence, with the times required to do each of these jobs being independent random variables with a mean of$50$ minutes and a standard deviation of$10$ minutes. M.J. has $20$jobs that he must do in sequence, with the times required to do each of these jobs being independent random variables with a mean of$52$ minutes and a standard deviation of $15$minutes.

$\left(a\right)$ Find the probability that A.J. finishes in less than $900$minutes.

$\left(b\right)$Find the probability that M.J. finishes in less than$900$ minutes.

$\left(c\right)$ Find the probability that A.J. finishes before M.J.

In Problem$8.7$, suppose that it takes a random time, uniformly distributed over$(0, .5)$, to replace a failed bulb. Approximate the probability that all bulbs have failed by time$550$.

It $X$is a gamma random variable with parameters$(n, 1)$, approximately how large must $n$be so that$P\left\{\left|\frac{X}{n}\right|-1>.01\right\}<.01?$

Civil engineers believe that W, the amount of weight (in units of $1000$pounds) that a certain span of a bridge can withstand without structural damage resulting, is normally distributed with a mean of $400$and standard deviation of$40$. Suppose that the weight (again, in units of $1000$pounds) of a car is a random variable with a mean of $3$and standard deviation$.3$. Approximately how many cars would have to be on the bridge span for the probability of structural damage to exceed$.1$?

Many people believe that the daily change in the price of a company’s stock on the stock market is a random variable with a mean of $0$and a variance of${\sigma }^2$. That is if Yn represents the price of the stock on the$n$ th day, then $Y_n = Y_{n-1} + X_n ,n \ge 1$where $X_1, X_2, ...$are independent and identically distributed random variables with mean $0$and variance${\sigma }_2$. Suppose that the stock’s price today is$100$. If${\sigma }_2= 1$, what can you say about the probability that the stock’s price will exceed $105$after$10$ days?

We have $100$components that we will put to use in a sequential fashion. That is, the component $1$is initially put in use, and upon failure, it is replaced by a component$2$, which is itself replaced upon failure by a component$3$, and so on. If the lifetime of component i is exponentially distributed with a mean $10 + i/10, i = 1, ... , 100$estimate the probability that the total life of all components will exceed$1200$. Now repeat when the life distribution of component i is uniformly distributed over$(0, 20 + i/5), i = 1, ... , 100$.

An insurance company has $10,000$automobile policyholders. The expected yearly claim per policyholder is$240$ a standard deviation of$800$. Approximate the probability that the total yearly claim exceeds $2.7$a million.

Redo Example$5b$ under the assumption that the number of man-woman pairs is (approximately) normally distributed. Does this seem like a reasonable supposition?

Repeat part $\left(a\right)$ of Problem$8.2$when it is known that the variance of a student’s test score is equal

to$25$.

If, $X$is a nonnegative random variable with a mean of$25$, what can be said about

$\left(a\right) E\left[X^3\right]?$

$\left(b\right) E[\surd X]?$

$\left(c\right) E[\log X]?$

$\left(d\right) E\left[e^{-X}\right]?$

Student scores on exams given by a certain instructor have mean 74 and standard deviation 14. This instructor is about to give two exams, one to a class of size 25 and the other to a class of size 64.

(a) Approximate the probability that the average test score in the class of size 25 exceeds 80.

(b) Repeat part (a) for the class of size 64.

(c) Approximate the probability that the average test score in the larger class exceeds that of the other class by more than 2.2 points.

(d) Approximate the probability that the average test score in the smaller class exceeds that of the other class.

by more than 2.2 points.

A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of these components must be in stock so that the probability that the system is in continual operation for the next 2000 hours is at least 0.95?

A.J. has 20 jobs that she must do in sequence, with the times required to do each of these jobs being independent random variables with mean 50 minutes and standard deviation 10 minutes. M.J. has 20 jobs that he must do in sequence, with the times required to do each of these jobs

being independent random variables with mean 52 minutes and standard deviation 15 minutes.

(a) Find the probability that A.J. finishes in less than 900 minutes.

(b) Find the probability that M.J. finishes in less than 900 minutes.

(c) Find the probability that A.J. finishes before M.J.

A lake contains 4 distinct types of fish. Suppose that each fish caught is equally likely to be any one of these types. Let Y denote the number of fish that need be caught to obtain at least one of each type.

(a) Give an interval (a, b) such that $P(a\le Y\le b)\ge 0.9$

(b) Using the one-sided Chebyshev inequality, how many fish need we plan on catching so as to be at least 90 percent certain of obtaining at least one of each type?

Compute the measurement signal-to-noise ratio that is$\left|\mu \right|/\sigma$, where $\mu = E\left[X\right]$ and ${\sigma }^2 = Var\left(X\right)$of the following random variables:

$\left(a\right)$ Poisson with mean$\lambda$;

$\left(b\right)$ binomial with parameters $n$and$p$;

$\left(c\right)$ geometric with mean$1/p$;

$\left(d\right)$ uniform over$(a, b)$;

$\left(e\right)$ exponential with mean$1/\lambda$;

$\left(f\right)$ normal with parameters$\mu , {\sigma }^2$.

Would the results of Example$5f$change be if the investor were allowed to divide her money and invest the fraction$\alpha , 0 <\alpha < 1,$ in the risky proposition and invest the remainder in the risk-free venture? Her return for such a split investment would be$R = \alpha X + (1 - \alpha )m$.

Let $X$be a non-negative random variable. Prove that

$E\left[X\right]\le {\left(E\left[X^2\right]\right)}^{1/2}\le {\left(E\left[X^3\right]\right)}^{1/3}\le \cdots$

It$X$ is a Poisson random variable with a mean$100$, then$PX>120$ is approximately

$\left(a\right) .02,$

$\left(b\right) .5$ or

$\left(c\right) .3?$

Let $X$be a Poisson random variable with a mean of$20$.

$\left(a\right)$Use the Markov inequality to obtain an upper bound$p=P(X\ge 26)$.

$\left(b\right)$Use the one-sided Chebyshev inequality to obtain an upper bound$p$.

$\left(c\right)$Use the Chernoff bound to obtain an upper bound$p$.

$\left(d\right)$ Approximate $p$by making use of the central limit theorem.

$\left(e\right)$Determine $p$by running an appropriate program.

It$X$ has, a mean$\mu$ and standard deviation$\sigma$, the ratio$r=\left|\mu \right|/\sigma$ is called the measurement signal-to-noise ratio$X$. The idea is that $X$can be expressed as$X = \mu + (X - \mu )$, $\mu$representing the signal and $X - \mu$the noise. If we define$\left|\right(X - \mu )/\mu |=D$ it as the relative deviation $X$ from its signal (or mean)$\mu$, show that for$\alpha > 0$,

$P\{D\le \alpha \}\ge 1-\frac{1}{r^2{\alpha }^2}$.

P{D≤α}≥1−1r2α2

Let$Zn$, $n \ge 1$ be a sequence of random variables and$c$ a constant such that for each

$\epsilon >0,P|Zn-c|>\epsilon \to 0$as$n\to q$. Show that for any bounded continuous function$g$,

$E\left[g\right(Zn\left)\right]\to g\left(c\right)$ as$n\to q$.

It $X$has a variance${\sigma }^2$, then σ, the positive square root of the variance, is called the standard deviation. It $X$has to mean $\mu$and standard deviation$\sigma$, to show that$P\left\{\right|X-\mu |\ge k\sigma \}\le \frac{1}{k^2}.$

Let $f\left(x\right)$be a continuous function defined for$0 \le x \le 1$. Consider the functions

$B_n\left(x\right) = \sum _{k=0}^n f\left(\frac{k}{n}\right) \left(\frac{n}{k}\right) x^k{(1 - x)}^{n-k}$ (called Bernstein polynomials) and prove that

$\underset{n\to \infty }{\lim } B_n\left(x\right) = f\left(x\right)$.

Hint: Let $X1, X2, ...$be independent Bernoulli random variables with mean$x$. Show that

$B_n\left(x\right) = E\left[f\left(\frac{x_1+...+x_n}{n}\right)\right]$

and then use Theoretical Exercise$8.4$.

Since it can be shown that the convergence of $B_n\left(x\right)$to $f\left(x\right)$is uniform$x$, the preceding reasoning provides a probabilistic proof of the famous Weierstrass theorem of analysis, which states that any continuous function on a closed interval can be approximated arbitrarily closely by a polynomial.

Suppose that a fair die is rolled $100$times. Let $X_i$be the value obtained on the $i$th roll. Compute an approximation for$P\left\{\prod _1^{100}{X}_i\le a^{100}\right\} 1 < a < 6$.

$\left(a\right)$Let $X$be a discrete random variable whose possible values are$1, 2, ...$. If $P[X=k]$is nonincreasing$k = 1, 2, ...$, prove that

$P(X=k)\le 2\frac{E\left[X\right]}{k^2}$

$\left(b\right)$Let $X$be a non-negative continuous random variable having a nonincreasing density function. Show that$f\left(x\right)\le 2\frac{E\left[X\right]}{x^2}$ for all$x>0$.

Compute the measurement signal-to-noise ratio that is, |μ|/σ, where μ = E[X] and σ2 = Var(X) of the

following random variables:

(a) Poisson with mean λ;

(b) binomial with parameters n and p;

(c) geometric with mean 1/p;

(d) uniform over (a, b);

(e) exponential with mean 1/λ;

(f) normal with parameters μ, σ2.

Suppose that the number of units produced daily at factory A is a random variable with mean $20$ and standard deviation $3$ and the number produced at factory B is a random variable with mean $18$ and standard deviation of $6$. Assuming independence, derive an upper bound for the probability that more units are produced today at factory B than at factory A. 

If $E\left[X\right]=75 E\left[Y\right]=75 Var\left(X\right)=10$

$Var\left(Y\right)=12 Cov(X,Y)=-3$

give an upper bound to 

(a) $P\left\{\right|X-Y|>15\};$

(b) $P\{X>Y+15\};$

(c) $P\{Y>X+15\}$;

8.5 The amount of time that a certain type of component functions before failing is a random variable with probability density function

$f\left(x\right)=2x 0<x< 1$ 

Once the component fails, it is immediately replaced by
another one of the same type. If we let denote the life-time of the $i$th component to be put in use, then $S_n=\sum _{i=1}^n{X}_i$represents the time of the $n$th failure. The long-term rate at which failures occur, call it $r$, is defined by
$r=\underset{n\to \infty }{\lim }\frac{n}{S_n}$

Assuming that the random variables $X i, i \ge 1,$are independent, determine $r$.

8.6 . In Self-Test Problem $8.5$, how many components would one need to have on hand to be approximately $90$ percent certain that the stock would last at least $35$ days?

8.7. The servicing of a machine requires two separate steps, with the time needed for the first step being an exponential random variable with mean $.2$hour and the time for the second step being an independent exponential random variable with mean $.3$ hour. If a repair person has $20$ machines to service, approximate the probability that all the work can be completed in $8$ hours.

Explain why a gamma random variable with parameters$(t, \lambda )$ has an approximately normal distribution when$t$ is large.

Suppose a fair coin is tossed $1000$times. If the first $100$tosses all result in heads, what proportion of heads would you expect on the final$900$ tosses? Comment on the statement “The strong law of large numbers swamps but does not compensate.”

It $X$is a Poisson random variable with a mean$\lambda$, showing that for$i < \lambda$,

$P\{X\le i\}\le \frac{e^{-\lambda }(e\lambda {)}^i}{i^i}$

The Chernoff bound on a standard normal random variable$Z$ gives$P\{Z>a\}\le e^{-a^2/2},a>0$. Show, by considering the density$Z$, that the right side of the inequality can be reduced by the factor$2$. That is, show that

$P\{Z>a\}\le \frac{1}{2}{e}^{-a^2/2} a>0$

 Let $X$be a binomial random variable with parameters $n$and$p$. Show that, for$i>n p$,

$\left(a\right)$ the minimum $e^{-ti}E\left[e^{tX}\right]$occurs when $t$is such that$e^t=$$\frac{iq}{(n-i)p}$ where$q=1-p.$

$\left(b\right)$$P\{X\ge i\}\le \frac{n^n}{i^2(n-i{)}^{n-i}}{p}^i(1-p{)}^{n-i}$

Show that if $E\left[X\right]<0$and $\theta \ne 0$ is such that$E\left[e^{\theta X}\right]=1$, then$\theta >0$.

The number of automobiles sold weekly at a certain dealership is a random variable with an expected value of$16$. Give an upper bound to the probability that

$\left(a\right)$ next week’s sales exceed$18$;

$\left(b\right)$ next week’s sales exceed$25$.

 Let $X_1,X_2,\dots$be a sequence of independent and identically distributed random variables with distribution$F$, having a finite mean and variance. Whereas the central limit theorem states that the distribution of$\sum _{i=1}^n{X}_i$ approaches a normal distribution as $n$goes to infinity, it gives us no information about how large$n$ need to be before the normal becomes a good approximation. Whereas in most applications, the approximation yields good results whenever$n\ge 20$, and oftentimes for much smaller values of$n$, how large a value of $n$is needed depends on the distribution of$X_i$. Give an example of distribution $F$such that the distribution$\sum _{i=1}^{100}{X}_i$ is not close to a normal distribution.

Hint: Think Poisson.

Suppose in Problem $8.14$that the variance of the number of automobiles sold weekly is$9$.

$\left(a\right)$Give a lower bound to the probability that next week’s sales are between $10$and$22$, inclusively.

$\left(b\right)$ Give an upper bound to the probability that next week’s sales exceed$18$.

A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of these components must be in stock so that the probability that the system is in continual operation for the next 2000 hours is at least .95? 

A tobacco company claims that the amount of nicotine in one of its cigarettes is a random variable with a mean of $2.2$ mg and a standard deviation of $0.3$ mg. However, the average nicotine content of $100$ randomly chosen cigarettes was $3.1$mg. Approximate the probability that the average would have been as high as or higher than $3.1$ if the company’s claims were true 

Each of the batteries in a collection of $40$batteries is equally likely to be either a type A or a type B battery. Type A batteries last for an amount of time that has a mean of $50$and a standard deviation of $15$; type B batteries last for a mean of $30$ and a standard deviation of 6.

(a) Approximate the probability that the total life of all $40$ batteries exceeds $1700$ 

(b) Suppose it is known that $20$ of the batteries are type A and $20$are type B. Now approximate the probability that the total life of all $40$ batteries exceeds $1700.$