# Exponential rate with Poisson

#### Pennstater

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
$t_A$ 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 $P[t_A < t_{BC}]$

#### Pennstater

Would the equation be the same if they gave me a mean of 1/mu?

#### romsek

MHF Helper
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 $\lambda = \dfrac 1 \mu$

#### Pennstater

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
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?

#### Pennstater

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
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)