91Ó°ÊÓ

A coin is biased so that a head is three time:s as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice.

By investing in a particular stock, a person can make a profit in one year of \(\$ 4000\) with probability 0.3 or take a loss of \(\$ 1000\) with probability 0.7 . What is this person's expected gain?

Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the probabilities are \(0.22,0.36,0.28,\) and \(0.14 .\) respectively, that she will be able to sell it, for a profit of \(\$ 250,\) sell it for a profit of \(\$ 150,\) break even, or sell it for a loss of \(\$ 150\). What is her expected profit?

A private pilot wishes to insure his airplane for \(\$ 200.000 .\) The insurance company estimates that a total loss may occur with probability \(0.002,\) a \(50 \%\) loss with probability \(0.01,\) and a \(25 \%\) loss with probability 0.1. Ignoring all other partial losses, what premium should the insurance company charge each year to realize an average profit of \(\$ 500 ?\)

If a dealer's profit, in units of \(\$ 5000,\) on a new automobile can bo looked upon as a random variable \(X\) having the density function $$f(x)=\left\\{\begin{array}{l}2(1-x) \\\0\end{array}\right.,$$ $$0

The hospital period, in days, for patients following treatment for a certain typo of kidney disorder is a random variable \(Y=X+4,\) where \(X\) has the density function $$f(x)=\left\\{\begin{array}{l}\frac{32}{(x-4)^{3}} \\\0\end{array}\right.,$$ $$x>0$$ Find the average number of days that a person is hospitalized following treatment for this disorder.

Suppose that the probabilities are 0.4. 0.3,0.2 , and \(0.1,\) respectively, that 0,1,2 . or 3 power failures will strike a certain subdivision in any given year. Find the mean and variance of the random variable \(X\) representing the number of power failures striking this subdivision.

Given a random variable \(A\), with standard deviation \(\sigma_{X}\) and a random variable \(Y=a+b X\). show that if \(b<0,\) the correlation coefficient \(\rho_{X Y}=-1,\) and if \(b>0, \rho_{X Y}=1\).

Seventy new jobs are opening up at an automobile manufacturing plant, but 1000 applicants show up for the 70 positions. To select the best 70 from among the applicants, the company gives a test that covers mechanical skill, manual dexterity, and mathematical ability. The mean grade on this test turns out to be \(60,\) and the scores have a standard deviation 6 . Can a person who has an 84 score count on getting one of the jobs? [Hint Use Chebyshev's theorem.] Assume that the distribution is symmetric about the mean.

4.67 A random variable \(X\) has a mean \(p=10\) and variance \(\sigma^{2}=4\). Using Chebyshev's theorem, find (a) \(P(X-10 \mid \geq 3)\) (b) \(\mathrm{P}(|\mathrm{X}-10|<3)\) (c) \(P(b