Let X equal the number of alpha particle emissions of carbon

Let X equal the number of alpha particle emissions of carbon-14 per second that are counted by a Geiger counter. Assume that X has a Poisson distribution with mean 8. Let W be the time in seconds before the second count is made. Determine P(W<=0.5) and P(W>0.75).

Solution

I\'ll show you the first one:
First the number of emissions in the first 1/2 sec. is N which has a Poisson distribution with mean 4:
Now the events { W<=0.5 } and {N >2} are equivalent and therefore,
P(W<=0.5) = P( N > 2) = 1 - P( N=0) -P(N=1) - P(N=2) = .9084.
The second case is solved by observing that P(W >.75) = P(M < 2) where M ~ Poisson (mean = 6).