A radioactive source emits certain particles with a Poisson distribution. The probability of no particle emissions during an hour of observation is $0.4$. What is the probability that the first recorded emission happens within the first $40$ to $60$ minutes?
I'm given that $\lambda = \dfrac{ \ln 2.5}{ 60 }$
I'm also given the answer of $0.1429$ but I don't know how to get it.
Best Answer
First get the Poisson distribution's rate parameter $\lambda$ from $\exp (-60\lambda)=0.4$ (taking one minute as our unit of time). The time to the first recorded emission is then exponentially distributed with this rate parameter, so the probability of it taking $40$ to $60$ minutes is $\exp (-40\lambda)-\exp (-60\lambda)=0.4^{2/3}-0.4$, which as required is $0.1429$.