Segment 8. Some Standard Distributions

Contents

Watch this segment

(Don't worry, what you see statically below is not the beginning of the segment. Press the play button to start at the beginning.)

{{#widget:Iframe |url=http://www.youtube.com/v/EDYDC7iNGTg&hd=1 |width=800 |height=625 |border=0 }}

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?)