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4. Let X be a random variable with cdf F and a strictly unimodal continuous pdf f (i.e. the pdf only has 1 mode or 1 global maximum, the local maximum is the global maximum). Assume that the mode of f (3:) is attained at :c 2 33M. Prove the following statements. (a) Let b > 0 be given and dene 9(a) : F(a + b) F(a). Show that 9(a) is maximized at a : a' which satises a' + b) u") = 0. Hints: i. You need to show that 9(a) is increasing when a a'. ii. Consider 2 cases: What happens when a,a + b 59M? (b) Suppose that for a given, a*, 5* are chosen such that F(a*+b*)F(a*) : 1a and f(a*+b*)f(a*) : 0. Prove that [a*, a\" +b*] is the shortest interval of the form [(3, 0+d] satisfying F(c+d) F(c) : 104. In visualizing this problem, it may be useful to understand what F(a + b) F (a) means. Perform the following. i. First, let a*,b* be such that F(a* + 53*) F(a*) = 1 a and f(a* + b*) f(a*) = 0. Consider another set of points a.' , 5' such that F (a'+b') F (a' ) : 1a but for which we have no information about a' + b') f (a' ) : 0. ([a', a." + b'] is our competing interval). Explain that our goal is to show that b\"