Probability: Poisson distribution

Question

Molecules are emitted in a rate which distribute poisson(0.5). (0.5 molecule per/second rate).

calculate the probabilities:

1. At least one molecule will be emitted in a certain second.
2. More than 3 molecules will be emitted within 5 seconds.

thanks.

Answer 1

The number of molecules emitted in a second has Poisson distribution, parameter $0.5$.

The probability of at least one molecule is $1$ minus the probability of $0$ molecules.

The probability of $0$ molecules is $e^{-0.5}$.

For the $5$ second problem, use the fact that the number of molecules in a $5$ second interval has Poisson distribution with parameter (mean) equal to $(0.5)(5)$.

To find the probability of more than $3$ molecules, first find the probability of $3$ or fewer molecules. This will involve adding four terms.