516hw8 - Solution to Homework 8 Section 4.2 1 Let X denote...

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Solution to Homework 8 Section 4.2 1. Let X denote the lifetime of an atom. Then X has an exponential distribution with log 2 λ = (as half life is 1). a) 5 ( 5) 1/32 P X e λ - = = . b) ( ) .1 .1 (log10) / 3.32 t P X t e t t λ λ - = = = = . c) Assume that lifetimes of individual atoms are independent. Then the number t N of atoms remaining after t years has a Binomial (1024, t e λ - ) distribution. Thus, ( ) 1 1024 1 10 t t E N e t λ - = = = . d) p=P(atom remains after 10 years)=P(X>10)=1/1024, let N be number of atoms remain after 10 years. N has binomial(1024, 1/1024) distribution which is approximately Poisson(1). So 1 ( 0) 0.3679 P N e - = = = . 2. a) Let X be the lifetime of an atom. Then X follows exponential distribution with log 2 λ = (as half life is 1). 2 ln 2 20 10 ( ) 10 t P X t e - = = 59.8 60 t = b) Let N be the number of atoms remain after t centuries. Then N has Bin( 20 10 , p). p=P(atom remains after t centuries)= ln 2 ( ) t P X t e - = . We have ( 0) .5 P N = = . Using Poisson approximation: 20 ln 2 20 10 10 2 t t np e μ - - = = = P(N=0)=0.5 20 10 2 66.97 67 t e e t μ - - - = = 4. a) Let W be the lifetime of a component. Then W
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