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homework4 - EL 630 Homework 4 1 A fair die is rolled until...

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1 EL 630: Homework 4 1. A fair die is rolled until a two appears. Let the random variable X be the number of rolls required. Find the probability distribution function ) ( x F X of the r.v X . 2. A fair coin is rolled 5,000 times. What is the probability that the total number of heads is between 2,225 and 2,525? 3. Players X and Y roll two dice alternatively starting with X . The player that rolls eleven wins. Find the probability p that X wins. 4. For the probability density function (p.d.f), < < - = , , 0 , 1 1 |, | ) ( elsewhere x x x f X find . 2 1 6 1 - X P Also find the distribution function of X . 5. In a restaurant known for its unusual service, the time X , in minutes, that a customer has to wait before he gets the attention of a waiter is specified by the following distribution function = . 20 , 1 , 20 10 , 20 / , 10 2 , 2 / 1 , 2 1 , 4 / , 1 0 , ) 2 / ( ) ( 2 x x x x x x x x x F X (a) Sketch ). ( x F X (b) Compute and sketch the p.d.f ) ( x f X of X . Verify that the area under ) ( x f X is unity. (c) What is the probability that a customer will have to wait (i) at least 10 minutes (ii) less than 5 minutes (iii) between 5 and 10 minutes (iv) exactly 1 minute.
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