Friday, 21 February 2014

Probability distributions

The chart I showed you, with the distributions we are studying marked in yellow, can be downloaded here. It is from the paper Univariate Distribution Relationships, L. M. Leemis and J. T. McQueston. In that paper you can find explanations of the various relationships that are indicated by the arrows.

Here also is a simpler chart about Relationships between probability distributions. This one shows some relations that the above chart misses out, such as how the binomial can come from the Poisson as $X \mid X+Y$ (see Examples sheet 2, #5).

Wikipedia has a nice List of probability distributions.

In 16.5 of the notes I mention that if $X$ and $Y$ are i.i.d. exponential r.vs then $X/(X+Y)$ is a uniformly distributed r.v. You will prove this in Examples sheet 4, #8. A general method will be given in Lecture 20. But it can also be done this way:

$P(X/(X+Y) \leq t)=P(X(1-t)/t\leq Y)=\int_{x=0}^\infty\left(\int_{y=x(1-t)/t}^{\infty}\lambda e^{-\lambda  y}dy\right)\lambda e^{-\lambda x}dx = t$

which is the c.d.f. of $U[0,1]$.