Exponential rate with Poisson

Apr 2019
4
0
State
A post office has two clerks. After three people, A, B, and C, enter simultaneously, A andB go directly to the clerks and C waits until either A or B leaves before she begins service. What is the probability that A is still in the post office after B and C have left when the service times are independent and exponentially distributed with rate λ?
I've completed the other parts, but I'm stuck on this. Any advice?
 

romsek

MHF Helper
Nov 2013
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$t_A$ is exponential w/parameter $\lambda$

$t_{BC}$ is exponential w/parameter $2\lambda$

$t_A,~t_{BC}$ are jointly exponential $P[t_A,t_{BC}] = P[t_A]P[t_{BC}]$

Find $P[t_A < t_{BC}]$
 
Apr 2019
4
0
State
Would the equation be the same if they gave me a mean of 1/mu?
 

romsek

MHF Helper
Nov 2013
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Would the equation be the same if they gave me a mean of 1/mu?
I don't understand what you're asking.

If you're told the exponential distribution has mean $\mu$ then it's parameter is $\lambda = \dfrac 1 \mu$
 
Apr 2019
4
0
State
A previous question gives just the mean and the current gives a new rate of just lamda. I'm not sure how to use the parameters or where they go
 

romsek

MHF Helper
Nov 2013
6,647
2,994
California
A previous question gives just the mean and the current gives a new rate of just lamda. I'm not sure how to use the parameters or where they go
can you post the question(s) in it's entirety?
 
Apr 2019
4
0
State
Yes.

Consider a post office with two clerks. Three people, A, B, and C, enter at the
same time. A and B go directly to the clerks, and C waits until either A or B
leaves before he begins service. What’s the probability that A is still in the post
office after the other two have left, i.e., P(A > B + C), when. . . ?

A)The service times are i.i.d. Exp(µ)

B) service times are independent and exponentially distributed with rate λ?
 

romsek

MHF Helper
Nov 2013
6,647
2,994
California
Yes.

Consider a post office with two clerks. Three people, A, B, and C, enter at the
same time. A and B go directly to the clerks, and C waits until either A or B
leaves before he begins service. What’s the probability that A is still in the post
office after the other two have left, i.e., P(A > B + C), when. . . ?

A)The service times are i.i.d. Exp(µ)

B) service times are independent and exponentially distributed with rate λ?
I don't really get what this problem is trying to show. Maybe I don't understand the terminology

The service times are i.i.d. Exp(µ)
what is $\exp(\mu)$ supposed to mean? Exponentially distributed with mean $\mu$?

If so it's just exponentially distributed with rate $\lambda = \dfrac{1}{\mu}$

So if you solve A) just substitute $\mu=\dfrac 1 \lambda$ into your solution and it will be the solution to B)