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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 12 - Nov 20 2017

Some notes


Exercise: see the textbook’s treatment of discrete mixtures of continuous r.v.’s.

Example 1

Let XExp(λ) and YExp(μ), εBernoulli(12). They are all independent.

let Z=X if ε=0 and Z=Y if ε=1.

Exercise: See book on `continuous mixtures’.

Example 2

XPoisson(λ). Assume λ itself is random. λΓ(m,θ).

Easy to say, X is Poisson conditional on λ, but what is the unconditional distribution of X?

Answer is in the book. We just want to compute P(X=k) for k=0,1,2,.

Bivariate Normal

Let XN(μ,σ2). We know we can represent X as X=μ+σZ where ZN(0,1).

More generally, let X1,X2 be bivariate normal. It turns out that we can represent X2 using X1 and an independent component ε2 like this:

X2=a+bX1+ε2

where a&b are constants. ε2 is a normal r.v. independent of X , with E(ε2).

We would like to compute a and b and Var(ε2). All we know is

Var(X1)=σ12 Var(X2)=σ22 Corr(X,Y)=ρ

To simplify, assume σ1=σ2, then we know, from linear prediction, that b=ρ. Then for Var(ε2):

Var(X2)=1=Var(bX1)+Var(ε2)=Var(ε2)=1ρ2

Also note: by taking expectation of the whole model,

μ2=a+bμ1+0

Therefore, a=μ2ρμ2.


Going back to our work with densities for the multivariate normal. We find the following density for the pair X=(X1,X2):

f(x1,x2)=constexp(12d(x122ρx1x2+x22))

where d=1ρ2, and const =12πd. Notice that

d=1ρ2=det(1ρρ1)

Also note: the expression

Q(x)=Q(x1,x2)=(x122ρx1x2+x22)/(2d)

is the quadratic form’’, which we encountered as the term 1/2xT[cov(X)]1x. Go back and check this is true.

When ρ=0, const 12π.

f(x1,x2)=12πexp(12(x12+x22))=12πexp(12x12)exp(12x22)

This proves the independence (ρ=0).



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