Solution Found!
Refer to Exercise 4.168.a If U is the time until the
Chapter 4, Problem 170SE(choose chapter or problem)
Problem 170SE
Refer to Exercise 4.168.
a If U is the time until the second arrival, show that U has a gamma density function with α = 2 and β = 1/λ.
b Show that the time until the kth arrival has a gamma density with α = k and β = 1/λ.
Reference
The number of arrivals N at a supermarket checkout counter in the time interval from 0 to t follows a Poisson distribution with mean λt. Let T denote the length of time until the first arrival. Find the density function for T . [Note: P(T > t0) = P(N = 0 at t = t0).]
Questions & Answers
QUESTION:
Problem 170SE
Refer to Exercise 4.168.
a If U is the time until the second arrival, show that U has a gamma density function with α = 2 and β = 1/λ.
b Show that the time until the kth arrival has a gamma density with α = k and β = 1/λ.
Reference
The number of arrivals N at a supermarket checkout counter in the time interval from 0 to t follows a Poisson distribution with mean λt. Let T denote the length of time until the first arrival. Find the density function for T . [Note: P(T > t0) = P(N = 0 at t = t0).]
ANSWER:
Solution:
Step 1 of 2:
a). To show that U has a gamma density function with .
If U is the time until the second arrival.
The second arrival will occur after time after time ‘t’ if either one arrival has occurred in (0, t) or no arrival has occurred in (0, t).
=
So, F(t) = 1- P(U > 1)
F(t)
, t > 0
This is a gamma density function with and
.