Calculus Applied to Probability and Statistics

Calculus Applied to Probability and Statistics - P Calculus...

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P.1 Continuous Random Variables and Histograms P.2 Probability Density Functions: Uniform, Exponential, Normal, and Beta P.3 Mean, Median, Variance, and Standard Deviation KEY CONCEPTS REVIEW EXERCISES CASE STUDY TECHNOLOGY GUIDES Calculus Applied to Probability and Statistics P 1 Case Study: Creating a Family Trust You are a financial planning consultant at a neighborhood bank. A 22-year-old client asks you the following question: “I would like to set up my own insurance policy by opening a trust account into which I can make monthly payments starting now, so that upon my death or my ninety-fifth birthday—whichever comes sooner—the trust can be expected to be worth $500,000. How much should I invest each month?” Assuming a 5% rate of return on investments, how should you respond?
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Introduction To answer the question on the previous page, we must know something about the prob- ability of the client’s dying at various ages. There are so many possible ages to consider (particularly because we should consider the possibilities month by month) that it would be easier to treat his age at death as a continuous variable, one that can take on any real value (between 22 and 95 in this case). The mathematics needed to do probability and statistics with continuous variables is calculus. The material on statistics in this chapter is accessible to any reader with a “common-sense” knowledge of probability, but it also supplements any previous study you may have made of probability and statistics without using calculus. Continuous Random Variables and Histograms Suppose that you have purchased stock in Colossal Conglomerate, Inc., and each day you note the closing price of the stock. The result each day is a real number X (the closing price of the stock) in the unbounded interval [0, +∞ ). Or, suppose that you time several people running a 50-meter dash. The result for each runner is a real number X , the race time in seconds. In both cases, the value of X is somewhat random. Moreover, X can take on essentially any real value in some interval, rather than, say, just integer values. For this reason, we refer to X as a continuous random variable . Here is the formal definition. 2 Chapter P Calculus Applied to Probability and Statistics P.1 Continuous Random Variable A random variable is a function X that assigns to each possible outcome in an experiment a real number. If X may assume any value in some given interval I (the interval may be bounded or unbounded), it is called a continuous random variable. If it can assume only a number of separated values, it is called a discrete random variable. Quick Examples 1. Roll a die and take X to be the number on the uppermost face. Then X is a discrete random variable with possible values 1, 2, 3, 4, 5, and 6.
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