(Device lifetime) A satellite orbiting the Earth contains a device whose lifetime X (in years) has the exponential distribution with parameter λ = 0.5. (a) Show how the cumulative distribution function F(t) is obtained by integrating the density. (b) Directly from the CDF, calculate the probabilities of the events A = {X ≤ 1}, B = {X > 5}, and C {5 < X < 6}. Give the results with at least 6 decimals. Explain in words what this events are. = (c) From the numerical results in (b), and using the definition of conditional probability, calculate P(C|B). Compare to P(A) and explain. (d) Consider a very short interval of h = 0.01 years. If the device has lasted to a certain point in time, what is the probability that it breaks during the next 0.01 years? Compare your numerical result to the value of Ah, and explain why A is called the rate parameter.
(Device lifetime) A satellite orbiting the Earth contains a device whose lifetime X (in years) has the exponential distribution with parameter λ = 0.5. (a) Show how the cumulative distribution function F(t) is obtained by integrating the density. (b) Directly from the CDF, calculate the probabilities of the events A = {X ≤ 1}, B = {X > 5}, and C {5 < X < 6}. Give the results with at least 6 decimals. Explain in words what this events are. = (c) From the numerical results in (b), and using the definition of conditional probability, calculate P(C|B). Compare to P(A) and explain. (d) Consider a very short interval of h = 0.01 years. If the device has lasted to a certain point in time, what is the probability that it breaks during the next 0.01 years? Compare your numerical result to the value of Ah, and explain why A is called the rate parameter.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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Transcribed Image Text:(Device lifetime) A satellite orbiting the Earth contains a device whose lifetime X (in
years) has the exponential distribution with parameter λ = 0.5.
(a) Show how the cumulative distribution function F(t) is obtained by integrating the density.
(b) Directly from the CDF, calculate the probabilities of the events A = {X ≤ 1}, B = {X >
5}, and C = {5 < X ≤ 6}. Give the results with at least 6 decimals. Explain in words
what this events are.
(c) From the numerical results in (b), and using the definition of conditional probability,
calculate P(C|B). Compare to P(A) and explain.
=
(d) Consider a very short interval of h 0.01 years. If the device has lasted to a certain point
in time, what is the probability that it breaks during the next 0.01 years? Compare your
numerical result to the value of Xh, and explain why A is called the rate parameter.
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