Michael Corral: Vector Calculus

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128 CHAPTER 3. MULTIPLE INTEGRALS 3.7 Application: Probability and Expected Value In this section we will briefly discuss some applications of multiple integrals in the field of probability theory. In particular we will see ways in which multiple integrals can be used to calculate probabilities and expected values. Probability Suppose that you have a standard six-sided (fair) die, and you let a variable X represent the value rolled. Then the probability of rolling a 3, written as P(X=3), is 1 6 , since there are six sides on the die and each one is equally likely to be rolled, and hence in particular the 3 has a one out of six chance of being rolled. Likewise the probability of rolling at most a 3, written as P(X≤ 3), is 3 6 = 1 2 , since of thesix numbers on the die, there are three equally likely numbers (1, 2, and 3) that are less than or equal to 3. Note that P(X≤ 3)=P(X= 1)+P(X= 2)+P(X= 3). We call X a discrete random variable on the sample space (or probabilityspace)Ωconsisting of all possible outcomes. In our case,Ω={1,2,3,4,5,6}. An event A is a subset of the sample space. For example, in the case of the die, the event X≤ 3 is the set{1,2,3}. Now let X be a variable representing a random real number in the interval (0,1). Note that the set of all real numbers between 0 and 1 is not a discrete (or countable) set of values, i.e. it can not be put into a one-to-one correspondence with the set of positive integers. 3 In this case, for any real number x in (0,1), it makes no sense to consider P(X=x) since it must be 0 (why?). Instead, we consider the probability P(X≤x), which is given by P(X≤x)= x. The reasoning is this: the interval (0,1) has length 1, and for x in (0,1) the interval (0,x) has length x. So since X represents a random number in (0,1), and hence is uniformly distributed over (0,1), then P(X≤x)= length of (0,x) length of (0,1) = x 1 = x. We call X a continuous random variable on the sample spaceΩ=(0,1). An event A is a subset of the sample space. For example, in our case the event X≤x is the set (0,x). In the case of a discrete random variable, we saw how the probability of an event wasthesumoftheprobabilitiesoftheindividualoutcomescomprisingthatevent(e.g. P(X≤ 3)=P(X= 1)+P(X= 2)+P(X= 3) in the die example). For a continuous random variable, the probability of an event will instead be the integral of a function, which we will now describe. Let X be a continuous real-valued random variable on a sample spaceΩin. For 3 For a proof see p. 9-10 in KAMKE, E., Theory of Sets, New York: Dover, 1950.
3.7 Application: Probability and Expected Value 129 simplicity, letΩ=(a,b). Define the distribution function F of X as F(x)=P(X≤x), for−∞< x
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About the author: Michael Corral is
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iv Preface matter, anyone can make
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vi Contents 4.4 Surface Integrals a
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2 CHAPTER 1. VECTORS IN EUCLIDEAN S
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178 CHAPTER 4. LINE AND SURFACE INT
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188 Bibliography Pogorelov, A.V., A
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190 Appendix A: Answers and Hints t
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Appendix B We will prove the right-
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Appendix C 3D Graphing with Gnuplot
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198 Appendix C: 3D Graphing with Gn
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200 Appendix C: 3D Graphing with Gn
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202 GNU Free Documentation License
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Index Symbols D....................
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212 Index iterated integral .......
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Michael Corral: Vector Calculus

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