hw5 solutions

# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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36-220 Engineering Stats, Fall 2005 Homework 5 Solutions Due: October 5, 2005 Non-Devore Exercise Let T be an exponential random variable that represents the time until a radioactive atom decays and λ be the rate of decay. a) We formulate the following probability statement – ”the probability that any given atom of plutonium has decayed after 24,110 years is exactly 0.5” – as Pr ( T 24110) = 0 . 5. In terms of the CDF of an exponential distribution, F : F (24110) = 1 - e - λ · 24110 = 0 . 5 Solving for λ , we have λ = - ln . 5 24110 b) We want to ﬁnd Pr ( T 1). Again write the probability in terms of F . Pr ( T 1) = F (1) = 1 - e - λ 1 - (1 - λ ) = λ = - ln . 5 24110 c) Let N be the number of atoms that decay within a year. Note N is a binomial random variable. Why? Let’s check that the assumptions are satisﬁed. Assumptions of Binomial Random Variable: 1. Fixed number of trials . Check. We have 10 · 2 . 5 · 10 24 atoms. 2. Each trial has 2 possible outcomes . Check. Either decay within a year or not. 3. Trials are independent . Check. Cosma gave you that assumption. 4. Probability of success is constant . It’s just the answer to part b), λ . Since the above assumptions hold, N is a binomial random variable with parameters n = 10 · 2 . 5 · 10 24 and p = λ , where n is the number of trials and p is probability of success (i.e. decay within a year). 1

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Finally, we can solve the problem: the expected number of atoms that decay within a year is just the expected value of N , which is np = 10 · 2 . 5 · 10 24 · λ 7 . 187 · 10 20 . Exercise 4.5 a) Recall that for a continuous function f to be a probability density function(pdf), summing f ( x ) over all x should give 1, i.e. R -∞ f ( x ) dx = 1. Using this fact, we can ﬁnd k , the unknown. 1 = Z -∞ f ( x ) dx = Z -∞ kx 2 dx = Z 2 0 kx 2 dx since f ( x ) = 0 for x < 0 and x > 2 = ± kx 3 3 ² ² ² ² 2 0 = 8 k 3 k = 3 8 The hint for b) to d) requires us to recall that Pr ( a X b ) = Z b a f ( x ) dx = F ( b ) - F ( a ) So to ﬁnd the probability of the form: Pr ( a X b ), we just need to ﬁnd the CDF of X and simply plug and chug a and b into F . Now let’s ﬁnd F .
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hw5 solutions - 36-220 Engineering Stats Fall 2005 Homework...

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