# Eleisha's Segment 8: Some Standard Distributions

** To Calculate **

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 {\rm Gamma}(\alpha,\beta) has a single mode at (\alpha-1)/\beta when \alpha \ge 1.

3. Show that the limiting case of the Student distribution as \nu\rightarrow\infty 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(\beta) random variable conditioned on its being greater than some given value?)