11 bivariate

Hadley Wickham

Stat310Bivariate distributions

Monday, 16 February 2009

1. Recap

2. Transformations, the cdf and the uniform distribution

3. Introduction to bivariate distributions

4. Properties of pdf. Marginal pdfs & expectation

5. Feedback


X ~ Exponential(θ). Y = log(X). What is fY(y)?

X ~ Uniform(0, 10). Y = X2. What is fY(y)?

Recap


Theorem 3.5-1

IF

Y ~ Uniform(0, 1)

F a cdf

X = F-1(Y)

THEN

X has cdf F(x)(Assume F strictly increasing for simplicity of proof, not needed in general)


Theorem 3.5-2

IF

X has cdf F

Y = F(X)

THEN

Y ~ Uniform(0, 1)(Assume F strictly increasing for simplicity of proof, not needed in general)


http://www.johndcook.com/distribution_chart.html






Bivariate random variables

Bivariate = two variables


Bivariate rv

Previously dealt with single random variables at a time.

Now we’re going to look at two (probably related) at a time

New tool: multivariate calculus


f(x, y) =116

! 2 < x, y < 2

What is:

• P(X < 0) ?

• P(X < 0 and Y < 0) ?

• P(Y > 1) ?

• P(X > Y) ?

• P(X2 + Y2 < 1)

Draw diagrams and use your intuition

What would you call this distribution?


f(x, y) = c a < x, y < b

How could we work out c?Is this a pdf?


f(x, y) ! 0 "x, y

! !

R2f(x, y) = 1


S = {(x, y) : f(x, y) > 0}Called the support or sample space


What is the bivariate cdf going to look like?


What is the bivariate cdf going to look like?

F (x, y) =! x

!"

! y

!"f(u, v)dvdu


Your turn

F(x, y) = c(x2 + y2) -1 < x, y < 1

What is c?

What is f(x, y)?


fX(x) =!

Rf(x, y)dy

fY (y) =!

Rf(x, y)dx

Marginal distribution of X

Marginal distribution of Y


Demo


Feedback


11 bivariate

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