By Mark A. Pinsky, Samuel Karlin

ISBN-10: 0123814162

ISBN-13: 9780123814166

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**Extra info for An Introduction to Stochastic Modeling, Fourth Edition**

**Example text**

2 The Exponential Distribution This exercise is designed to foster intuition about the exponential distribution, as well as to provide practice in algebraic and calculus manipulations relevant to stochastic modeling. Let X0 and X1 be independent exponentially distributed random variables with respective parameters λ0 and λ1 , so that Pr{Xi > t} = e−λi t for t ≥ 0, i = 0, 1. Let N= 0 if X0 ≤ X1 , 1 if X1 ≤ X0 ; U = min{X0 , X1 } = XN ; M = 1 − N; V = max{X0 , X1 } = XM ; 38 An Introduction to Stochastic Modeling and W = V − U = |X0 − X1 |.

It plays a role in the class of discrete distributions that parallels in some sense that of the normal distribution in the continuous class. The Poisson distribution occurs often in natural phenomena, for powerful and convincing reasons (the law of rare events, see later in this section). At the same time, the Poisson distribution has many elegant and surprising mathematical properties that make analysis a pleasure. The Poisson distribution with parameter λ > 0 has the probability mass function p(k) = λk e−λ k!

E[g(X)|Y = y] = E[g(X)] if X and Y are independent. 5. E[g(X)h(Y)|Y = y] = h(y)E[g(X)|Y = y]. 11) h(y)E[g(X)|Y = y] pY (y) 6. E[g(X)h(Y)] = y = E{h(Y)E[g(X)|Y]}. 14) E[g(X)|Y = y] pY (y) = E{E[g(X)|Y]}. 1 I roll a six-sided die and observe the number N on the uppermost face. I then toss a fair coin N times and observe X, the total number of heads to appear. What is the probability that N = 3 and X = 2? What is the probability that X = 5? What is E[X], the expected number of heads to appear? 2 Four nickels and six dimes are tossed, and the total number N of heads is observed.

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

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