Random Variables - Derived Distributions and PDF Calculations
Learn how to calculate the PDF of a random variable using derived distributions, step-by-step approaches, and examples. Understand linear and monotonic cases for PDF calculations. Explore scenarios like archer shooting to apply the concepts practically.
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Tutorial 8: Further Topics on Random Variables 1 Weiwen LIU wwliu@cuhk.edu.hk March 20, 2017 1
Derived distributions Given ? = ?(?) of a continuous random variable ? and PDF of ?, how to calculate the PDF of ?? Two step approach 2
Calculation of PDF of ? = ?(?) 1. Calculate the CDF ?? of ? using the formula ??? = ? ? ? ? = ??(?)?? {?|?(?) ?} 2. Differentiate ??to obtain the PDF ?? of ?: ??? =??? ? . ?? 3
Calculation of PDF of ? = ?(?) Linear case: The PDF of ? = ?? + ? in terms of the PDF ?. ??? =??? 1 ? ? ? ? = |?|?? ?? Monotonic Case: Suppose that ?is monotonic and that for some function and all ?in the range of ? we have ? = ? ? if and only if ? = (?). Assume that is differentiable. ? ??(?) ??? = ?? (?) 4
Example 1 Let X be a random variable with PDF ??. Find the PDF of the random variable ? = |?|, 1 3,?? 2 < ? 1, 0, ?? ??????; (a) when ??? = (b) when ??? = 2? 2?,?? ? > 0, 0, ?? ??????; (c) for general ??(?). 5
Example 1 Since ? = ? , you can visualize the PDF for any given ? as ??(?) = ??? + ?? ? , 0, Also note that since ? = |?|, ? 0. ?? ? 0 ?? ? < 0 6
Example 1 1 3,?? 2 < ? 1, 0, ?? ??????; (a) Since ??? = So, ??(?) for 1 ? 0 gets added to ??(?) for 0 ? 1: 2 3,?? 0 ? < 1, 1 3,?? 1 ? < 2, 0, ?? ??????; ??? = 7
Example 1 (b) Here we are told ? > 0. So there are no negative values of ? that need to be considered. Thus f?y = ??? = 2? 2?,?? ? > 0, 0, ?? ??????. (c) As explained in the beginning, ??(?) = ??(?) + ??( ?). 8
Example 2: archer shooting Two archers shoot at a target. The distance of each shot from the center of the target is uniformly distributed from 0 to 1, independent of the other shot. Question: What is the PDF of the distance of the winning shot from the center? 9
Example 2: archer shooting Let ? and ? be the distances from the center of the first and second shots, respectively. ?: the distance of the winning shot: ? = min ?,? . Since ? and ? are uniformly distributed over 0,1 , we have, for all ? 0,1 , ? ? ? = ? ? ? = 1 ?. 10
Example 2: archer shooting Thus, using the independence of ? and ?, we have for all ? 0,1 , ??? = 1 ? min{?,?} ? = 1 ? ? ? ? ? ? = 1 1 ?2. Differentiating, we obtain ??? = 2(1 ?), 0, if 0 ? 1. otherwise. 11
Covariance The covariance of two random variables ? and ? are defined as cov ?,? = E E ? E E ? ? ? ? . Alternatively cov ?,? = E E ?? E E ? E E ? . 12
Correlation Coefficient For any random variable ?, ? with nonzero variances, the correlation coefficient ? ?,? of them is defined as cov(?,?) ?(?,?) = . var ? var(?) It may be viewed as a normalized version of the covariance cov(?,?). Recall cov ?,? = var ? . It s easily verified that 1 ?(?,?) 1 13
Example 3 A restaurant serves three fixed-price dinners costing $12, $15, and $20. For a randomly selected couple dinning at this restaurant, let ? = ? ? ???? ?? ? ? ??? ? ?????? and ? = ? ? ???? ?? ? ? ????? ? ??????. If the joint PMF of ? and ? is assumed to be, what is cov ?,? ? 14
Example 3 Cov ?,? = ?[??] ? ? ?[?] = 276.7 15.9 17.45 = 0.755 15
