Scaling a Poisson distribution

Question

Is it possible to scale a Poisson distribution and receive the same result. Lets say that I have bridge A. On average 10 cars drive over bridge A per hour, thus if I want to calculate the probability that at most 4 cars drove over bridge A after a given hour I would take the Poisson CDF with lambda 10, in R ppois(4,10), which is roughly 3%. However, lets say instead I would like to see the probability that 2 cars drove over the bridge in 30 minutes instead, same methodology as above i.e. ppois(2,5) which gives roughly 12.5%.

I would initially think that you would be able to scale, but thinking about it, does this happen as with fewer instances there is a larger chance that relatively speaking more events deviate?

Answer 1

The answer to your question is twofold.

1. Clarifying cdf: The cumulative distribution function (cdf) is a probability of a series of events. In R, ppois(k1,rt) is the dpois(k,rt) summed over all k<=k1. Here, r is the rate of the event per unit time t. Therefore, cdf is a sum of probabilities.

Hence, probability that two cars drove over the bridge in half an hour represented by the probability mass function dpois(2,5) (not the cdf ppois()).

2. Sum of Poisson distributed random variables: First consider the probability that exactly 4 cars pass through a bridge where on average 10 cars pass. This is represented by dpois(4,10). When split into half hour intervals, the average rate drops to 5 per half hour. However, the two half hour intervals can combine in several combinations to create the 4 cars in one hour: 0,4 1,3 2,2 3,1 4,0 respectively in fist half and the second half hour. Therefore, the probabilities for all these half-hour events combine to equate the former one hour event. dpois(4,10)=dpois(0,5)*dpois(4,5)+dpois(1,5)*dpois(3,5)+dpois(2,5)*dpois(2,5)+dpois(3,5)*dpois(1,5)+dpois(4,5)*dpois(0,5).