# Segment 8. Some Standard Distributions

## Contents

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### Problems

#### 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.

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?)