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Probability and Statistics

Instructions: Do your work on this question and answer sheet. Please print or write legibly, and, as always, be complete but succinct. Record your answer and your supporting work in the designated space. Explain your method of solution and be sure to label clearly any and all var-iables and symbols. I am not a mind reader, no more than you are an automaton that produces only equations and symbols. There is a single problem, which is worth a total of 100 points. Attach to these sheets the two Excel plots requested in (d). Please staple together these sheets and the Excel plots.1. (100 points) Exercise 3.10, page 91 in the textA rental agency, which leases heavy equipment by the day, has found that one expensive piece of equipment is leased, on the average, only one day in five. If rental on one day is in-dependent of rental on any other day, find the probability distribution of Y, the number of days between a pair of rentals..(a) (5 points) What is the range space XR, which is to ask what are the values of Y?Answer The range space YR=(b) (10 points) Find the probability distribution of Y, representing it by a formula.1Answer (5 points) ()(),YPyypyyR===∀∈Solution1 If you are unable to determine the probability distribution, then, if you request, I will tell you what it is. Although you would thereby forfeit the 10 points that part (b) is worth, you would not risk forfeiting potentially many points in subsequent parts that depend on knowledge of the probability distribution.2(c) (10 points) Verify that your (proposed) distribution satisfies the two defining properrties of a discrete probability distribution. Hint: In the verification of one of them, use the ap-propriate formula for the sum of a geometric series. This and other useful formulas are gath-ered in the document entitled "Formulas of the Variance and of Sums of Geometric Series".Solution(d) (15 points) Use Excel to plot the graph of the probability distribution of Y, at least over the range of the first sixteen values in the range space of Y. Plot on a second graph the distri-bution function of Y. Recall that, for any random variable Y, the (cumulative) distribution function of Y, denoted by ()()()for YFyFyPYyy==≤-∞<<∞. The distribution func-tion accumulates the probabilities up through a given y and corresponds to the cumulative (relative) frequency in a grouped frequency distribution. The correlate of the distribution function, which is important in actuarial studies, is the survival function of Y, denoted by ()()()for YSySyPYyy==>-∞<<∞. If Y denotes the (future) lifetime of the members of a population, then S(y) denotes the probability that an individual will live beyond y (years, say), that is, will survive beyond y. Observe that ()()()()11SyPYyPYyFy=>=-≤=-3Plot the survival function together with its correlate, the distribution function.Attach to this homework assignment the two plots (one of the probability distribution func-tion, the second displaying both the distribution and survival functions).Include on both plots appropriate chart and axis (horizontal and vertical) titles.(e) (5 points) Describe the shape of the probability distribution of Y. (The shape of the cu-mulative distribution and survival functions of Y is characteristic of them.)Answser(f) (5 points) Does Y have a mode? If it does, then identify it (them).Answer(g) (15 points) Using the cumululative distribution function and linear interpolation, deter-mine the median of Y to three decimal places.Answer (5 points) x=?Solution4(h) (20 points) What is the "average" number of days between pairs of rentals? The question more precisely phrased is: what is the mean or expected value E(Y) of Y? Hint: Use one of the formulas of the sum of a geometric series that is in the document entitled "Formulas of the Variance and of Sums of Geometric Series".Answer (5 points) ()YEYμ==Solution (15 points)5(i) (5 points) Observe how the mean, median, and mode compare, and relate this to the shape of the probability distribution of Y that you described in (e).Answer(j) (10 points) Using the formula in (3) in the document entitled "Formulas of the Variance and of Sums of Geometric Series" and, in that same document, a formula of the sum of a ge-ometric series, calculate the variance of (the given) Y.Answer (3 points) ()2yVYσ==Solution 

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