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Chapter 1. Introduction > 1.6. Useful Functions, Integrals, and Sums - Pg. 42

42 An Introduction to Stochastic Modeling 1.5.5 Show that E[W ] = 0 2 2y[1 - F W (y)]dy for a nonnegative random variable W. 1.5.6 Determine the upper tail probabilities Pr{V > t} and mean E[V] for a random variable V having the exponential density f V (v) = 0 e -v for v < 0, for v 0, where is a fixed positive parameter. 1.5.7 Let X 1 , X 2 , . . . , X n be independent random variables that are exponentially dis- tributed with respective parameters 1 , 2 , . . . , n . Identify the distribution of the minimum V = min{X 1 , X 2 , . . . , X n }. Hint: For any real number v, the event {V > v} is equivalent to {X 1 > v, X 2 > v, . . . , X n > v}. 1.5.8 Let U 1 , U 2 , . . . , U n be independent uniformly distributed random variables on the unit interval [0, 1]. Define the minimum V n = min{U 1 , U 2 , . . . , U n }. (a) Show that Pr{V n > v} = (1 - v) n for 0 v 1.