STT 861 Theory of Prob and STT I Lecture Note - 12
2017-11-20
Examples on continuous distirbution conditional on discrete distribution; bivariate normal distribution. Not much for today.
Portal to all the other notes
- Lecture 01 - 2017.09.06
- Lecture 02 - 2017.09.13
- Lecture 03 - 2017.09.20
- Lecture 04 - 2017.09.27
- Lecture 05 - 2017.10.04
- Lecture 06 - 2017.10.11
- Lecture 07 - 2017.10.18
- Lecture 08 - 2017.10.25
- Lecture 09 - 2017.11.01
- Lecture 10 - 2017.11.08
- Lecture 11 - 2017.11.15
- Lecture 12 - 2017.11.20 -> This post
- Lecture 13 - 2017.11.29
- Lecture 14 - 2017.12.06
Lecture 12 - Nov 20 2017
Some notes
- Exponential distribution and geometry distribution are the only distributions which have the memoryless property (continuous and discrete).
- Geometry is not scalable. It only takes integer values. is not a geometry distribution.
Exercise: see the textbook’s treatment of discrete mixtures of continuous r.v.’s.
Example 1
Let and , . They are all independent.
let if and if .
Exercise: See book on `continuous mixtures’.
Example 2
. Assume itself is random. .
Easy to say, is Poisson conditional on , but what is the unconditional distribution of ?
Answer is in the book. We just want to compute for .
Bivariate Normal
Let . We know we can represent as where .
More generally, let be bivariate normal. It turns out that we can represent using and an independent component like this:
where are constants. is a normal r.v. independent of , with .
We would like to compute and and . All we know is
To simplify, assume , then we know, from linear prediction, that . Then for :
Also note: by taking expectation of the whole model,
Therefore, .
Going back to our work with densities for the multivariate normal. We find the following density for the pair :
where , and const . Notice that
Also note: the expression
is the quadratic form’’, which we encountered as the term . Go back and check this is true.
When , const .
This proves the independence ().
- ← Older-STT 861 Theory of Prob and STT I Lecture Note - 11
- Things to Do After Installing Ubuntu-Newer →