Question

Suppose that the number of customers that enter a bank in an hour is a Poisson random variable, and suppose that P(X=0)=0.05. Determine the mean and variance of X.

Answer 1

To determine the mean and variance of a Poisson distribution given P(X=0)=0.05, one solves the equation e^-λ=0.05 for λ, which will then be equal to both the mean and variance.

Step-by-step explanation:

The student is asking about a scenario where the number of customers entering a bank in an hour follows a Poisson distribution. Given that P(X=0)=0.05, this information can be used to find the mean (also the variance for a Poisson distribution) because in a Poisson distribution, the probability P(X=k) is given by (e-λλk)/k! where λ is the mean and variance of the distribution. Since P(X=0) relates directly to the mean λ through the formula e-λ, one can set e-λ = 0.05 and solve for λ. After computing the mean, determining the variance is straightforward, since for a Poisson distribution mean = variance.