Eleisha's Segment 8: Some Standard Distributions

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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?)