Finding P(X= x) when x is a non-integer value in a Poisson distribution

Question

I am having some issues finding a resource to assist me with this. We have a Poisson distribution with on average sees 5 trains roll through its station every 1 hour. Let the random variable X be the time in hours between arrivals of consecutive trains. Find the P(X = 1/3).

I suppose I could break it down into minutes instead, then at the end of the problem work my way back up to hours but otherwise I am unable to use the general formula for solving probability.

Answer 1

In a Poisson process, two different kinds of random variables are involved:

• The number of arrivals in a specified time interval. That is a discrete random variable whose values are in the set $\{0,1,2,3,\ldots\}.$ This has a Poisson distribution.
• The amount of time between one arrival and the next. This is a continuous random variable whose value is not generally an integer. This does NOT have a Poisson distribution. It has an exponential distribution.

The probability that the time is exactly $1/3$ hour is $0$.

The probability that the time is at least $1/3$ hour is the same as the probability that the number of trains in that $1/3$ hour is $0$. The expected number of trains in $1/3$ hour is $5/3.$ The probability that the number of trains during that time is $0$ is therefore $\dfrac{(5/3)^0 e^{-5/3}}{0!} = e^{-5/3}.$