1. Stat310 Bivariate distributions
Hadley Wickham
Monday, 16 February 2009
2. 1. Recap
2. Transformations, the cdf and
the uniform distribution
3. Introduction to bivariate distributions
4. Properties of pdf. Marginal pdfs &
expectation
5. Feedback
Monday, 16 February 2009
3. Recap
X ~ Exponential(θ). Y = log(X).
What is fY(y)?
X ~ Uniform(0, 10). Y = X2.
What is fY(y)?
Monday, 16 February 2009
4. 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)
Monday, 16 February 2009
5. 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)
Monday, 16 February 2009
7. Bivariate random
variables
Bivariate = two variables
Monday, 16 February 2009
8. 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
Monday, 16 February 2009
11. 1
f (x, y) = − 2 < x, y < 2
16
What would you call
What is:
this distribution?
• P(X < 0) ?
• Draw diagrams and
P(X < 0 and Y < 0) ?
use your intuition
• P(Y > 1) ?
• P(X > Y) ?
• P(X2 + Y2 < 1)
Monday, 16 February 2009
12. f (x, y) = c a < x, y < b
Is this a pdf?
How could we work out c?
Monday, 16 February 2009
13. f (x, y) ≥ 0 ∀x, y
f (x, y) = 1
R2
Monday, 16 February 2009
14. S = {(x, y) : f (x, y) > 0}
Called the support or
sample space
Monday, 16 February 2009
15. What is the
bivariate cdf going
to look like?
Monday, 16 February 2009
16. What is the
bivariate cdf going
to look like?
x y
F (x, y) = f (u, v)dvdu
−∞ −∞
Monday, 16 February 2009
17. Your turn
F(x, y) = c(x 2 + y 2) -1 < x, y < 1
What is c?
What is f(x, y)?
Monday, 16 February 2009
18. Marginal distribution of X
fX (x) = f (x, y)dy
R
Marginal distribution of Y
fY (y) = f (x, y)dx
R
Monday, 16 February 2009
