Chapter 8

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

From past experience, a professor knows that the test score of a student taking her final examination is a random variable with mean 75 (a) Give an upper bound for the probability that a student's test score will exceed \(85 .\) (b) Suppose, in addition, that the professor knows that the variance of a student's test score is equal to \(25 .\) What can be said about the probability that a student will score between 65 and 85? (c) How many students would have to take the examination to ensure with probability at least. 9 that the class average would be within 5 of \(75 ?\) Do not use the central limit theorem.

Let \(X_{1}, \ldots, X_{20}\) be independent Poisson random variables with mean 1. (a) Use the Markov inequality to obtain a bound on $$P\left\\{\sum_{1}^{20} X_{i}>15\right\\}$$ (b) Use the central limit theorem to approximate $$P\left\\{\sum_{1}^{20} X_{i}>15\right\\}$$.

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 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.

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 mean 400 and standard deviation \(40 .\) Suppose that the weight (again, in units of 1000 pounds) of a car is a random variable with mean 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 of price of a company's stock on the stock market is a random variable with mean 0 and variance \(\sigma^{2}\). That is, if \(Y_{n}\) represents the price of the stock on the \(n\) th day, then $$Y_{n}=Y_{n-1}+X_{n} \quad n \geq 1$$ where \(X_{1}, X_{2}, \ldots\) 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 in use in a sequential fashion. That is, component 1 is initially put in use, and upon failure, it is replaced by component \(2,\) which is itself replaced upon failure by component \(3,\) and so on. If the lifetime of component \(i\) is exponentially distributed with mean \(10+i / 10, i=1, \ldots, 100,\) 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 \((0,20+i / 5), i=1, \ldots, 100\).

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.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 \(P\\{a \leq Y \leq b\\}\) \(\geq .90\) (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?

Let \(X\) be a nonnegative random variable. Prove that $$E[X] \leq\left(E\left[X^{2}\right]\right)^{1 / 2} \leq\left(E\left[X^{3}\right]\right)^{1 / 3} \leq \dots$$

Would the results of Example \(5 \mathrm{f}\) change 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 Poisson random variable with mean \(20 .\) (a) Use the Markov inequality to obtain an upper bound on $$p=P\\{X \geq 26\\}$$ (b) Use the one-sided Chebyshev inequality to obtain am upper bound on \(p\). (c) Use the Chernoff bound to obtain an upper bounc on \(p\). (d) Approximate \(p\) by making use of the central limi theorem. (e) Determine \(p\) by running an appropriate program.

If \(X\) is a Poisson random variable with mean 100 then \(P\\{X>120\\}\) is approximately (a) .02 (b) .5 or (c).3?