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homework05 - Probability for Engineering Applications...

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Probability for Engineering Applications Assignment #5, due at the beginning of class on Thursday, February 23 th , 2006 1. (3 pts) Are the following functions proper cdf’s? Justify your answers. Note that u ( x ) represents the unit step function. a. F X ( x ) = (1 - e - 2 x ) u ( x ). b. F X ( x ) = 0 . 5 u ( x ) + 0 . 1 u ( x - 1) + 0 . 3 u ( x - 3). c. F X ( x ) = 0 . 3 u ( x ) + 0 . 3 u ( x - 1) - 0 . 2 u ( x - 2) + 0 . 6 u ( x - 3). 2. (2 pts) (Textbook exercise 3.5) A dart is thrown onto a square b units wide. Assume that the dart is equally likely to fall anywhere in the square. Let the random variable Z be given by the sum of the x and y coordinates of the point where the dart lands. a. Describe the sample space of Z , S Z . b. Find the region in the square corresponding to the event { Z z } for -∞ < z < . (Show it using a figure). c. Find P [ Z z ]. 3. (2 pts) (Textbook exercise 3.13) A communication system with three links is shown in Figure 1. Each link may fail or may not fail independently of the other links in a given interval of time. We define the probability that link i fails by P [ F i ], i = 1 , 2 , 3 and have P [ F 1 ] = 0 . 01 P [ F 2 ] = 0 . 10 P [ F 3 ] = 0 . 20 Let M i denote the capacity of link
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