91Ó°ÊÓ

Suppose a life insurance company sells a \(\$ 50,000\), five-year term policy to a twenty-five-year-old woman. At the beginning of each year the woman is alive, the company collects a premium of \(\$ P\). The probability that the woman dies and the company pays the \(\$ 50,000\) is given in the table below. So, for example, in Year 3 , the company loses \(\$ 50,000-\$ P\) with probability \(0.00054\) and gains \(\$ P\) with probability \(1-0.00054=0.99946\). If the company expects to make \(\$ 1000\) on this policy, what should \(P\) be? \begin{tabular}{cc} \hline Year & Probability of Payoff \\ \hline 1 & \(0.00051\) \\ 2 & \(0.00052\) \\ 3 & \(0.00054\) \\ 4 & \(0.00056\) \\ 5 & \(0.00059\) \\ \hline \end{tabular}

Records show that 642 new students have just entered a certain Florida school district. Of those 642 , a total of 125 are not adequately vaccinated. The district's physician has scheduled a day for students to receive whatever shots they might need. On any given day, though, \(12 \%\) of the district's students are likely to be absent. How many new students, then, can be expected to remain inadequately vaccinated?

Calculate \(E(Y)\) for the following pdfs: (a) \(f_{Y}(y)=3(1-y)^{2}, 0 \leq y \leq 1\) (b) \(f_{Y}(y)=4 y e^{-2 y}, y \geq 0\) (c) \(f_{Y}(y)= \begin{cases}\frac{3}{4}, & 0 \leq y \leq 1 \\ \frac{1}{4}, & 2 \leq y \leq 3 \\ 0, & \text { elsewhere }\end{cases}\) (d) \(f_{Y}(y)=\sin y, \quad 0 \leq y \leq \frac{\pi}{2}\)

Show that the expected value associated with the exponential distribution, \(f_{Y}(y)=\lambda e^{-\lambda y}, y>0\), is \(1 / \lambda\), where \(\lambda\) is a positive constant.

Suppose that fifteen observations are chosen at random from the pdf \(f_{Y}(y)=3 y^{2}, 0 \leq y \leq 1\). Let \(X\) denote the number that lie in the interval \(\left(\frac{1}{2}, 1\right)\). Find \(E(X)\).

A city has 74,806 registered automobiles. Each is required to display a bumper decal showing that the owner paid an annual wheel tax of \(\$ 50\). By law, new decals need to be purchased during the month of the owner's birthday. How much wheel tax revenue can the city expect to receive in November?

A fair coin is tossed three times. Let the random variable \(X\) denote the total number of heads that appear times the number of heads that appear on the first and third tosses. Find \(E(X)\).

For the St. Petersburg problem (Example 3.5.5), find the expected payoff if (a) the amounts won are \(c^{k}\) instead of \(2^{k}\), where \(0<\) \(c<2\). (b) the amounts won are \(\log 2^{k}\). [This was a modification suggested by D. Bernoulli (a nephew of James Bernoulli) to take into account the decreasing marginal utility of money \(-\) the more you have, the less useful a bit more is.]

Suppose that two evenly matched teams are playing in the World Series. On the average, how many games will be played? (The winner is the first team to get four victories.) Assume that each game is an independent event.

Find the median for each of the following pdfs: (a) \(f_{Y}(y)=(\theta+1) y^{\theta}, 0 \leq y \leq 1\), where \(\theta>0\) (b) \(f_{Y}(y)=y+\frac{1}{2}, 0 \leq y \leq 1\)