Consider the Poisson process of radioactive particle hits in Example 5.7.8. Suppose that the rate of the Poisson process is unknown and has the

Consider the Poisson process of radioactive particle hits in Example 5.7.8. Suppose that the rate β of the Poisson process is unknown and has the gamma distribution with parameters α and γ. Let X be the number of particles that strike the target during t time units. Prove that the conditional distribution of β given X = x is a gamma distribution, and find the parameters of that gamma distribution.