1 STAT 511-2 Spring 2012 Lecture 14 Feb 10, 2012 Jun Xie Finish the content of cumulative distribution function in the last post. 4.2 Expected valuesDefinition The expected or mean value of a continuous rvX with pdf f(x) is x= E(X) =𝑥𝑓 𝑥 𝑑𝑥∞−∞. Proposition If X is a continuous rv with pdf f(x) and h(X) is any function of X, then E[h(X)] = h(X)= ℎ(𝑥)𝑓 𝑥 𝑑𝑥∞−∞. Example 11Two species are competing in a region for control of a limited amount of a certain resource. Let X= the proportion of the resource controlled by species 1 and suppose X has pdf of uniform on [0,1]. Then the species that controls the majority of this resource controls the amounth(X) = max (X, 1 –X). The expected amount controlled by the species having majority control is then E[h(X)] = 𝑚𝑎𝑥(𝑥, 1− 𝑥)𝑓 𝑥 𝑑𝑥∞−∞=1− 𝑥 𝑑𝑥1/20+𝑥𝑑𝑥11/2=12−12𝑥2|01/2+12𝑥2|1/21=34. Definition The variance of a continuous random variable X with pdf f(x) and mean value is
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