Introduction to Probability and Statistics for Engineers and Scientists
144 Chapter 5: Special Random Variables Because each package will, independently, have to be replaced with probability .005, it ...
5.1The Bernoulli and Binomial Random Variables 145 whereas the corresponding probability for a 3-component system is ( 3 2 ) p^2 ...
146 Chapter 5: Special Random Variables EXAMPLE 5.1d Suppose that 10 percent of the chips produced by a computer hardware manufa ...
5.1The Bernoulli and Binomial Random Variables 147 Var(X)= ∑n i= 1 Var(Xi) since theXiare independent =np(1−p) IfX 1 andX 2 are ...
148 Chapter 5: Special Random Variables Binomial Distribution Enter Value For p: Enter Value For n: Enter Value For i: .75 100 7 ...
5.2The Poisson Random Variable 149 0 .20 .16 .12 .08 .04 (^012345678910) P {X = i} 11 12 i FIGURE 5.3 The Poisson probability ma ...
150 Chapter 5: Special Random Variables Evaluating att=0 gives that E[X]=φ′(0)=λ Var(X)=φ′′(0)−(E[X])^2 =λ^2 +λ−λ^2 =λ Thus both ...
5.2The Poisson Random Variable 151 The number ofα-particles discharged in a fixed period of time from some radioactive particle ...
152 Chapter 5: Special Random Variables from past experience that, on the average, 3.2 suchα-particles are given off, what is a ...
5.2The Poisson Random Variable 153 The Poisson approximation result can be shown to be valid under even more general conditions ...
154 Chapter 5: Special Random Variables The last equality follows since, from Equation 5.2.2, E[Xi]=P{Xi= 1 }= 1 n ■ The Poisson ...
5.2The Poisson Random Variable 155 Now, given a total ofn+mevents, because each one of these events is independently type 1 with ...
156 Chapter 5: Special Random Variables Starting withP{X= 0 }=e−λ, we can use Equation 5.2.7 to successively compute P{X= 1 }=λP ...
5.3The Hypergeometric Random Variable 157 To compute the mean and variance of a hypergeometric random variable whose prob- abili ...
158 Chapter 5: Special Random Variables Also, fori<j, Cov(Xi,Xj)=E[XiXj]−E[Xi]E[Xj] Now, because bothXiandXjare Bernoulli (th ...
5.3The Hypergeometric Random Variable 159 throughout the region, a new catch of size, say,nis made. LetXdenote the number of mar ...
160 Chapter 5: Special Random Variables where the next-to-last equality used the fact thatX+Y is binomial with parameters (n+m,p ...
5.4The Uniform Random Variable 161 f(x) a b 1 b – a ab x FIGURE 5.5 Probabilities of a uniform random variable. is a subinterval ...
162 Chapter 5: Special Random Variables The mean of a uniform[α,β]random variable is E[X]= ∫β α x β−α dx = β^2 −α^2 2(β−α) = (β− ...
5.4The Uniform Random Variable 163 SOLUTION E[I]=E[I 0 (eaV−1)] =I 0 E[eaV− 1 ] =I 0 (E[eaV]−1) = 10 −^6 ∫ 3 1 e^5 x 1 2 dx− 10 ...
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