Scaling a Poisson distribution 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?
 A: The answer to your question is twofold.

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*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()).


*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).

