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Unformatted text preview: 2.8 Problems 79 including the play in which his friend Joe won a game. He conjectured that if p is the probability that Joe wins any game and the games are independent, then the PMF of N is given by p N ( n ) = p ( 1- p ) n- 1 n = 1 , 2 ,... a. Show that p N ( n ) is a proper PMF. b. Find the CDF of N . 2.14 The discrete random variable K has the following PMF: p K ( k ) = b k = 2 b k = 1 3 b k = 2 otherwise (a) What is the value of b ? (b) Determine the values of (i) P [ K < 2 ] , (ii) P [ K ≤ 2 ] , and (iii) P [ < K < 2 ] . (c) Determine the CDF of K . 2.15 A student got a summer job at a bank, and his assignment was to model the number of customers who arrive at the bank. The student observed that the number of customers K that arrive over a given hour had the PMF p K ( k ) = braceleftbigg λ k e- λ / k ! k = , 1 , 2 ,... otherwise (a) Show that p K ( k ) is a proper PMF. (b) What is P [ K > 1 ] ? (c) What is P [ 2 ≤ K ≤ 4 ] ? 2.16 Let X be the random variable that denotes the number of times we roll a fair die until the first time the number 5 appears. Find the probability that X = k . 2.17 The PMF of a random variable X is given by p X ( x ) = b λ x / x ! , x = , 1 , 2 ,... , where λ > 0. Find the numerical values for (a) P [ X = 1 ] , (b) P [ X > 3 ] . 2.18 A random variable K has the PMF p K ( k ) = parenleftbigg 5 k parenrightbigg ( . 1 ) k ( . 9 ) 5- k k = . 1 ,..., 5 80 Chapter 2 Random Variables Obtain the values of the following: (a) P [ K = 1 ] (b) P [ K ≥...
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- Fall '07