# HW1Sol - Homework 1 solution ^ 1 H is a binomial random...

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Homework 1 – solution 1. ˆ H is a binomial random variable with parameters n = 20 and p = 0 . 7. a) Its probability distribution is given by P ( ˆ H = i ) = 20 i · 0 . 7 i 0 . 3 20 - i for i = 0 , 1 , . . . , 20. b) E ( ˆ H ) = np = 14. c) Std( ˆ H ) = q np (1 - p ) = 2 . 05. 2. The lifetime of any given bulb is T N (10000 , 1000). a) P ( T > 12000) = 1 - F (12000) = 1 - Φ 12000 - 10000 1000 · = 1 - Φ(2) = 1 - 0 . 977 = 0 . 023. b) P ( T < 9000) = F (9000) = Φ 9000 - 10000 1000 · = Φ( - 1) = 0 . 159. c) P (replacing at least one bulb) = 1 - P (replacing no bulbs) = 1 - P (all T i > 5000) = 1 - P ( T > 5000) 100 = 1 - (1 - F (5000)) 100 = 1 - (1 - Φ( - 5)) 100 = 0 (up to three decimal points). d) P (all bulbs burining out) = P (all T i < 15000) = P ( T < 15000) 100 = ( F (15000)) 100 = (Φ(5)) 100 = 1 (up to three decimal points). 3. Let X be the (random) volume to be put in a vial. We know that X N ( μ, σ ) , with σ = 0 . 1 and μ unknown (to be determined). The requirement that the fraction of under-filled vials be 0.01 is expressed as P ( X < 2 . 5) = 0 . 01 . Standardizing, we obtain P Z < 2 . 5 - μ σ = 0 . 01 , from which it follows that

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• Spring '08
• Perevalov
• Probability distribution, Probability theory, Cumulative distribution function, Discrete probability distribution, dx

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