Assignment 4.docx - 3.61 Suppose that X has a Weibull...

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3.61 Suppose that X has a Weibull distribution with β = ¿ 0.2 and δ = ¿ 100 hours. Determine the mean and variance of X Mean: β = ¿ 0.2 ¿ δ = ¿ 100 E(X) = 100 τ ( 1 + 1 0.2 ¿ 100 τ (6) = 100 (5)! E(X) = 12,000 hours Variance: β = ¿ 0.2 ¿ δ = ¿ 100 V (X) = 100 2 τ 2 ( 1 + 2 0.2 ¿ -100 2 ¿ ¿ τ ( 1 + 1 0.2 ¿ ] 2 100 2 τ ( 11 ) – 100 2 [ τ ( 6 ) ¿ 2 (100 2 (10!)) – (100 2 (5!) 2 ) = 3.6144E + 10 hours 2 3.64 The life (in hours) of a computer processing unit (CPU) is modeled by a Weibull distribution with parameters β =3 and δ = ¿ 900 hours. (a) Determine the mean life of the CPU. E(X) = 900 τ ( 1 + 1 3 ¿ 900 τ (4/3) = 900 (0.8929799) E(X) = 803.68 hours (b) Determine the variance of the life of the CPU. V (X) = 900 2 τ 2 ( 1 + 2 3 ¿ -900 2 ¿ ¿ τ ( 1 + 1 3 ¿ ] 2
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900 2 τ ( 5 / 3 ) – 900 2 [ τ ( 4 / 3 ) ¿ 2 (900 2 (0.90274584)) – (900 2 (0.8929799) 2 ) = 85319.52 hours 2 (c) What is the probability that the CPU fails before 500 hours? P (X < 500) = F (500) 1 – exp (-(500/900) 3 )) 1 – exp (-0.1715) = 0.1576 3.68 Suppose that X has a gamma distribution with λ = 3 r = 6. Determine the mean and variance of X. Mean E (X) = r/ λ 6/3 = 2 Variance V (X) = r/ λ 2 6/(3) 2 = 0.667 3.69 Suppose that X has a gamma distribution with λ = 2.5 r = 3.2 .
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