Fall 2011 Homework 6: Question 5
Customers arrive at the Leftorium store according to a Poisson process with rate $\lambda$ customers per hour. The true value of $\lambda$ is unknown, so we treat it as a random variable (this is called a Bayesian approach). Suppose that our prior beliefs about $\lambda$ can be expressed as $\lambda$ ~ Expo(3). Let X be the number of customers who arrive at the Leftorium between 1 pm and 3 pm tomorrow. Given that X = 2 is observed, find the conditional PDF of $\lambda$ (this is known as the posterior density of $\lambda$).
Solution: Use Bayes' rule to get the posterior density of lambda. Using the law of total probability, integration by parts, and lotus, we ultimately get that this is the Gamma(3, 5) distribution.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."