Segment 8. Some Standard Distributions

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Problems

To Calculate

1. In Segment 6 (slide 8) we used the improper prior Failed to parse (unknown error): 1/r . Show that this is just a limiting case of a (completely proper) Lognormal prior.

2. Prove that $\rm Gamma$ has a single mode at Failed to parse (unknown error): (\alpha-1)/\beta when Failed to parse (unknown error): \alpha \ge 1 .

3. Show that the limiting case of the Student distribution as Failed to parse (unknown error): \nu\rightarrow\infty is the Normal distribution.