RecitationFiveSols - 1 − F T (30) = 31 − 30 60 60 −...

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Rensselaer Electrical, Computer, and Systems Engineering Department ECSE 4500 Probability for Engineering Applications Recitation 5 Solutions Wednesday February 24, 2010 Ans: For the Bernoulli random variable, we have P [ X =0]= p and P [ X =1]=1 p , q, so the probability mass function (PMF) is P X ( k )= p, k =0 , q, k =1 , 0 , else . Thus in terms of delta functions, we have equivalently f X ( x )= ( x )+ ( x 1) . For a binomial random variable X with parameters n and p ,wehavethePMF P X ( k )= ½ ¡ n k ¢ p k q n k , 0 k n, 0 , else. Thus an equivalent probability density would be f X ( x )= n X k =0 μ n k p k q n k δ ( x k ) . For a Poisson random variable N with parameter μ ,th ePMFi s P N ( n )= μ n n ! e μ , thus the corresponding pdf would be f N ( x )= X n =0 μ n n ! e μ δ ( x n ) . Ans: Let random variable T be the professor’s arrival time. We then have the density and distribution functions sketched below.
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Then for the given de f nitions of the events A and B ,weget P [ A ]= P [ T> 30] = 1 F T (30) , P [ B ]= P [ T 31] = F T (31) , and P [ AB ]= P [30 <T 31] = F T (31) F T (30) . We then have P [ B | A ]= P [ AB ] P [ A ] = F T (31) F T (30) 1 F
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Unformatted text preview: 1 − F T (30) = 31 − 30 60 60 − 30 60 = 1 30 , and P [ A | B ] = P [ AB ] P [ B ] = F T (31) − F T (30) F T (31) = 31 − 30 60 31 60 = 1 31 . Correction: Please substitute λ for μ in this problem (2 places), as λ is conventionally used for the rate parameter . Also assume λ is measured in hours. Ans: The pdf of the failure time X is given from Section 2.7 in general as f X ( t ) = α ( t ) exp( − Z t α ( τ ) dτ ) . In this case the rate is constant and α ( t ) = λ, thus f X ( t ) = λ exp( − λt ) . If A = { failure in 100 hours or less } , then P [ A ] = P [ X ≤ 100] = Z 100 λe − λt dt = 1 − e − 100 λ . In order for this probability to be less than or equal to . 05 , we need e − 100 λ ≥ . 95 or λ ≤ 1 100 ln 1 . 95 We get 5 . 13 × 10 − 4 2...
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This note was uploaded on 04/15/2010 for the course ECSE 4500 taught by Professor Woods during the Spring '08 term at Rensselaer Polytechnic Institute.

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RecitationFiveSols - 1 − F T (30) = 31 − 30 60 60 −...

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