Download An Introduction to Stochastic Modeling, Fourth Edition by Mark A. Pinsky, Samuel Karlin PDF

By Mark A. Pinsky, Samuel Karlin

ISBN-10: 0123814162

ISBN-13: 9780123814166

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12 Suppose that the telephone calls coming into a certain switchboard during a one-minute time interval follow a Poisson distribution with mean λ = 4. If the switchboard can handle at most 6 calls per minute, what is the probability that the switchboard will receive more calls than it can handle during a specified one-minute interval? 13 Suppose that a sample of 10 is taken from a day’s output of a machine that produces parts of which 5% are normally defective. If 100% of a day’s production is inspected whenever the sample of 10 gives 2 or more defective parts, then what is the probability that 100% of a day’s production will be inspected?

1 Evaluate the moment E eλZ , where λ is an arbitrary real number and Z is a random variable following a standard normal distribution, by integrating +∞ λZ E[e ] −∞ 1 2 eλz √ e−z /2 dz. 2π Hint: Complete the square − 2l z2 + λz = − 12 (z − λ)2 − λ2 and use the fact that +∞ −∞ 1 2 √ e−(z−λ) /2 dz = 1. 2 Let W be an exponentially distributed random variable with parameter θ and mean µ = 1/θ. (a) Determine Pr{W > µ}. (b) What is the mode of the distribution? 3 Let X and Y be independent random variables uniformly distributed over the interval θ − 21 , θ + 21 for some fixed θ.

Conditioned on M, the random variable X has a binomial distribution with parameters M and π . (a) Determine the marginal distribution for X. (b) Determine the covariance between X and Y = M − X. 2 A card is picked at random from N cards labeled 1, 2, . . , N, and the number that appears is X. A second card is picked at random from cards numbered 1, 2, . . , X and its number is Y. Determine the conditional distribution of X given Y = y, for y = 1, 2, . . 3 Let X and Y denote the respective outcomes when two fair dice are thrown.

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