Exercise 3 a satellite system consists of n

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EXERCISE 3 A satellite system consists of n components and functions on any given day if at least k of the n components function on that day. On a rainy day each of the components independently functions with probability p 1 , whereas on a dry day they independently function with probability p 2 . The probability of a rainy day is θ . a. Give an expression of the probability that the satellite system will function at any given day. b. Evaluate the above probability when n = 3, k = 1, p 1 = 0 . 85, p 2 = 0 . 90, and θ = 0 . 20. c. If the satellite system functioned yesterday what is the probabilty that it was a rainy day? Use n = 3, k = 1, p 1 = 0 . 85, p 2 = 0 . 90, and θ = 0 . 20. EXERCISE 4 Let X be a geometric random variable with probability of success p . a. Show that for a positive integer k , P ( X > k ) = (1 - p ) k . b. Show that P ( X > n + k - 1 | X > n - 1) = P ( X > k ) EXERCISE 5 Two boys play in the following way. They take turns shooting and stop when a basket is made. Player A goes first and has probability p 1 of making a basket on any throw. Player B , who shoots second, has probability p 2 of making a basket. The outcomes of succesive outcomes are assumed to be independent.
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