The discussion revolves around calculating the new emission rate of a radioactive source based on a change in the probability of emissions over a specified time period. The context includes the application of the Poisson distribution to model the emissions, with a focus on the mathematical reasoning involved in determining the new rate.
Participants do not appear to reach a consensus on the methods used for calculation, with differing approaches mentioned but no definitive agreement on a single method being preferred.
The discussion includes assumptions about the Poisson distribution and the specific parameters used in the calculations, but these assumptions are not explicitly stated or resolved.
araz said:A radioactive source emits particles at an average rate of 1 pe second. Assume that the number of emissions follows a Poisson distribution. The emission rate changes such that the probability of 0 or 1 emission in 4 seconds becomes 0.8. What is the new rate? Thanks.
Klaas van Aarsen said:Hi araz, welcome to MHB! ;)
We have:
$$P(\text{0 or 1 emission in 4 seconds})=0.8 \\ \implies
P(\text{0 emission in 4 seconds}) + P(\text{1 emission in 4 seconds}) = \frac{\lambda^0 e^{-\lambda}}{0!} + \frac{\lambda^1 e^{-\lambda}}{1!} = (1+\lambda)e^{-\lambda} = 0.8 \\ \implies
\lambda \approx 0.824
$$
So the average number of emissions in 4 seconds is $0.824$, which is $0.206$ per second.
araz said:Thank you Klaas. Did you use guess and check or numerical methods to get 0.824?
Regards