Question

A device that detects the presence of radon gas in a home counts the number of alpha particles that hit its detector over a defined period. In a particular home, the number of alpha particles detected per week averages 3 over a long period of time. Let X represent the number of alpha particles that will be detected next week.

a) Suggest a suitable probability model for the random variable X. Justify your answer and identify any assumptions you are making.

b) State the mean, variance and standard deviation of your model.

c) What is the probability that 5 alpha particles will be detected next week?

d) What is the probability that 12 alpha particles will be detected over the next three weeks?

Answer 1

a) Here,

X counts the number of alpha particles that hit its detector over a week, so we can assume

X ~ Poisson with mean = 3

The probability mass function of X is

b) The mean of X is = E(X) = 3

The variance of X is = Var(X) = 3

The standard deviation of X is =

c) The probability that 5 alpha particles will be detected next week

d) Let Y denotes the number of alpha particles will be detected over the next three weeks.

Y ~ Poisson(3*3) or Y ~ Poisson(9)

The probability mass function of Y is

The probability that 12 alpha particles will be detected over the next three weeks