1. In Segment 6 (slide 8) we used the improper prior 1 / r. Show that this is just a limiting case of a (completely proper) Lognormal prior.
2. Prove that Gamma(α,β) has a single mode at (α − 1) / β when .
3. Show that the limiting case of the Student distribution as is the Normal distribution.
To Think About 1. Suppose you have an algorithm that can compute a CDF, P(x). How would you design an algorithm to compute its inverse (see slide 9) x(P)?
2. The lifetime t of a radioactive nucleus (say Uranium 238) is distributed as the Exponential distribution. Do you know why? (Hint: What is the distribution of an Exponential(β) random variable conditioned on its being greater than some given value?)