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homework06 - c What is the probability of one or more...

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Probability for Engineering Applications Assignment #6, due at the beginning of class on Thursday, March 2 nd , 2006 1. (2 pt) (Textbook exercise 3.21) A random variable X has the pdf shown in Figure 1. a. Find the pdf f X ( x ). (Evaluate the constant c ). b. Find the cdf F X ( x ). c. Find b such that P [ | X | < b ] = 1 / 4. fx(x) x c 0 -a a Figure 1: Probability density function 2. (2 pt) (Similar to textbook exercise 3.29) Consider a binomial distribution with n tries where the probability of success on any given try is p . Let the even A be defined as A = { X > 4 } . a. Find F X ( x | A ) and f X ( x | A ). b. Let the conditional pmf of X be given by P [ X = x | A ]. Find the conditional pmf. 3. (2pts) Consider a router in the Internet where data packets arrive at the rate of 5 packets per second and it is known that the number of packets arriving in any interval of time is a Poisson random variable. a. Find the probability that no packets arrive in a half-second interval. b. What is the probability of one or less arrivals in a half-second interval?
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Unformatted text preview: c. What is the probability of one or more arrivals in a half-second interval? 1 4. (2 pts) Shims are manufactured to a nominal thickness of 5 mm. However, the stock from which they are cut is known to vary from this thickness, which from extensive measurements is modeled as a Gaussian random variable with m = 5 mm and σ = 0 . 05 mm. What is the probability that the thickness of a shim is less than 4.9 mm or more than 5.1 mm? 5. (2 pts) (Textbook exercise 3.42) The r th percentile, π ( r ), of a random variable X is deFned by P [ X ≤ π ( r )] = r/ 100. a. ±ind the 90%, 95% and 99% percentiles of the exponential random variable with pa-rameter λ . b. Repeat part a for the Gaussian random variable with parameters m = 0 and σ . 2...
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