How Is the Conditional PMF Calculated for Car Arrivals at a Toll Station?

[maibs89]

Suppose that cars traveling at night on a freeway arrive at a toll station according to a Poisson process with rate alpha @=5 per minute. If five cars arrived in [0,3] (that is, during a three minute period starting at midnight), what is the conditional pmf of the number of cars that arrived in [0,1] (during the first minute of the period)?

 

[Pere Callahan]

Welcome to PF

What do you mean by pmf? Probability mass function?

Anyways, You condition on the even that up t=3 five cars arrived. what do you know about the conditional distribution of the arrival times of these five cars?

Hint: It's a famous property of the Poisson process because of which it is sometimes referred to as the "most random" point process of all.

 

[Architeuthis Dux]

The conditional pmf of the number of cars that arrived in [0,1] would be calculated as follows:

P(X=x|Y=5) = (e^(-5)*(5^x))/x!

Where X represents the number of cars that arrived in the first minute and Y represents the total number of cars that arrived in the entire three minute period.

In this case, since Y=5, the conditional pmf would be:

P(X=x|Y=5) = (e^(-5)*(5^x))/x!

= (e^(-5)*(5^x))/x!

= (e^(-5)*(5^x))/x!

= (e^(-5)*(5^1))/1!

= (e^(-5)*5)/1

= 0.0337

Therefore, the conditional pmf of the number of cars that arrived in [0,1] during the first minute of the period is 0.0337 or approximately 3.37%. This means that there is a 3.37% chance that exactly one car arrived during the first minute given that five cars arrived in the entire three minute period.