A6_Ch6_poisson_process_solutions

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Jan 9, 2024

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STAT 251 In-class activity worksheet – Chapter 6 - Solutions Poisson Process – Relationship between Poisson and Exponential Random variables SOLUTIONS The following is the histogram of the 67 recurrence intervals (times between earthquake occurrences). The curve is the Exponential probability density function f(t) based on the estimated rate parameter 0.2403 (annual rate of occurrence). 1. Assuming an Exponential distribution for the recurrence intervals, use the estimated annual rate of occurrence of earthquakes (0.2403) to calculate the probability that a recurrence interval is shorter than 10 months. Let T be the recurrence interval. T~Exp( λ ) where λ = 0.2403 𝑃 (𝑇 < 10 12 = 0.833) = 𝐹 𝑇 ( 10 12 ) = 1 − 𝑒 −0.2403× 10 12 = 0.1815

2. Using the above histogram to answer the following questions. What proportion of the observed recurrence intervals are greater than 5 years (hint: observed proportion = areas of bars where area = density × bar width)? How does it compare to the theoretical value assuming an Exponential distribution for the recurrence intervals? Proportion = 0.06 + 0.06 + 0.02 + 0.03 + 0.02 + 0.05 + 0.02 + 0.02 = 0.28 The observed proportion is close to the theoretical probability 𝑃(𝑇 > 5) = 1 − 𝐹(5) = 𝑒 −0.2403×5 = 0.301 3. Using a Poisson random variable and the estimated annual rate of earthquake occurrences, calculate the probability that you observe no earthquakes in 5 years. Estimated value of λ = 0.2403 per 1 year Let X = number of earthquakes in 5 years X ~ Poisson (λ t = 0.2403*5 = 1.2015) 𝑃(? = 0) = 𝑒 −1.2015 × 1.2015 0 0! = 0.301 4. Compare your answer in Q3 with the value obtained in Q2(a). What do you notice? Can you explain why? The two give the same answer. This is expected because the event of having to wait for more than 5 years before the next earthquake is the same as the event of having no earthquakes in 5 years. 𝑃(𝑇 > 5) = 𝑃(? = 0) = 𝑒 −1.2015 5. An earthquake just occurred. A seismologist starts counting the number of earthquakes that occur since then (not including the one that has just occurred). Estimate the probability that he has to wait more than 30 years to observe the 10 th earthquake. The waiting time until the 10 th earthquake is at least 30 years. This implies that during the next 30 years, there can be at most 9 earthquakes. Let Y = number of earthquakes in 30 years Y ~ Poisson (λt = 0.2403*30 = 7.209) 𝑃(? < 10) = 𝑃(? = 0) + 𝑃(? = 1) + 𝑃(? = 2) + . . . + 𝑃(? = 9) = ∑ 𝑒 −7.209 × 7.209 𝑦 𝑦! 9 𝑦=0 = 0.8087

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