# Eleisha's Segment 8: Some Standard Distributions

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To Calculate

1. In Segment 6 (slide 8) we used the improper prior $\displaystyle 1/r$ . Show that this is just a limiting case of a (completely proper) Lognormal prior.

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

3. Show that the limiting case of the Student distribution as $\displaystyle \nu\rightarrow\infty$ is the Normal distribution.

1. Suppose you have an algorithm that can compute a CDF, $\displaystyle P(x)$ . How would you design an algorithm to compute its inverse (see slide 9) $\displaystyle x(P)?$
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($\displaystyle \beta$ ) random variable conditioned on its being greater than some given value?)