1. Let \( N \) be a Poisson process with intensity \( \lambda \), with \( N = \{ N(t) : t \geq 0 \} \) where \( N(t) \) denotes the number of arrivals in the interval \( (0, t] \). Let \( T_0, T_1, \ldots \) be given by \[ T_0 = 0, T_n = \inf \{ t : N(t) = n \}. \] Define the interarrival times \( X_1, X_2, \ldots \) by \[ X_n = T_n - T_{n-1} \] Show that the random variables \( X_1, X_2, \ldots \) are independent and each have exponential distribution with parameter \( \lambda \).
1. Let \( N \) be a Poisson process with intensity \( \lambda \), with \( N = \{ N(t) : t \geq 0 \} \) where \( N(t) \) denotes the number of arrivals in the interval \( (0, t] \). Let \( T_0, T_1, \ldots \) be given by \[ T_0 = 0, T_n = \inf \{ t : N(t) = n \}. \] Define the interarrival times \( X_1, X_2, \ldots \) by \[ X_n = T_n - T_{n-1} \] Show that the random variables \( X_1, X_2, \ldots \) are independent and each have exponential distribution with parameter \( \lambda \).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. Let \( N \) be a Poisson process with intensity \( \lambda \), with \( N = \{ N(t) : t \geq 0 \} \) where \( N(t) \) denotes the number of arrivals in the interval \( (0, t] \). Let \( T_0, T_1, \ldots \) be given by
\[ T_0 = 0, T_n = \inf \{ t : N(t) = n \}. \]
Define the interarrival times \( X_1, X_2, \ldots \) by
\[ X_n = T_n - T_{n-1} \]
Show that the random variables \( X_1, X_2, \ldots \) are independent and each have exponential distribution with parameter \( \lambda \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F17f09d59-7004-42bf-bea0-369d03c59713%2Fwwzlzmi_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let \( N \) be a Poisson process with intensity \( \lambda \), with \( N = \{ N(t) : t \geq 0 \} \) where \( N(t) \) denotes the number of arrivals in the interval \( (0, t] \). Let \( T_0, T_1, \ldots \) be given by
\[ T_0 = 0, T_n = \inf \{ t : N(t) = n \}. \]
Define the interarrival times \( X_1, X_2, \ldots \) by
\[ X_n = T_n - T_{n-1} \]
Show that the random variables \( X_1, X_2, \ldots \) are independent and each have exponential distribution with parameter \( \lambda \).
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