91Ó°ÊÓ

If \(X\) has variance \(\sigma^{2}\), then \(\sigma\), the positive square root of the variance, is called the standard deviation. If \(X\) has mean \(\mu\) and standard deviation \(\sigma\), show that $$ P\\{|X-\mu| \geq k \sigma\\} \leq \frac{1}{k^{2}} $$

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 \(85 .\) Suppose, in addition, the professor knows that the variance of a student's test score is equal to 25 . (b) 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.

Compute the measurement signal-to-noise ratio-that is, \(|\mu| / \sigma\) where \(\mu=E[X], \sigma^{2}=\operatorname{Var}(X)\) - of the following random variables: (a) Poisson with mean \(\lambda\); (b) binomial with parameters \(n\) and \(p\); (c) geometric with mean \(1 / p\); (d) uniform over \((a, b)\); (e) exponential with mean \(1 / \lambda\); (f) normal with parameters \(\mu, \sigma^{2}\).

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

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 repairperson has 20 machines to service, approximate the probability that all the work can be completed in 8 hours.

On each bet, a gambler loses 1 with probability \(.7\), loses 2 with probability .2, or wins 10 with probability .1. Approximate the probability that the gambler will be losing after his first 100 bets.

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 that "the strong law of large numbers swamps but does not compensate."

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 over \(2.2\) points. (d) Approximate the probability that the average test score in the smaller class exceeds that of the other class by over \(2.2\) points.