Unraveling Multimodality with Large Language Models.pdf
12 Bivariate
1. Stat310
Bivariate random variables
Hadley Wickham
Friday, 26 February 2010
2. Assessment
• Please pick up any homework you
haven’t got already.
• Will be grading tests tomorrow to get
back to you on Thursday
• Drop deadline is Feb 26 – if you are
thinking of dropping and would like an
interim grade, email me Friday morning
Friday, 26 February 2010
3. 1. Introduction to bivariate random
variables
2. The important bits of multivariate
calculus
3. Independence
Friday, 26 February 2010
4. Bivariate rv
Previously dealt with one random variable
at a time. Now we’re going to look at two
(probably related) at a time.
A random experiment where we measure
two things (not just one).
New tool: multivariate calculus
Friday, 26 February 2010
7. 1
f (x, y) = − 2 < x, y < 2
16
What is: What would you call
this distribution?
• P(X < 0) ?
• P(X < 0 and Y < 0) ? Draw diagrams and
• P(Y > 1) ? use your intuition
• P(X > Y) ?
• P(X2 + Y2 < 1)
Friday, 26 February 2010
8. f (x, y) = c a < x, y < b
Is this a pdf?
How could we work out c?
Friday, 26 February 2010
9. Your turn
Given what you know about univariate
pdfs and pmfs, guess the conditions that
a bivariate function must satisfy to be a
bivariate pdf/pmf.
Friday, 26 February 2010
10. pdf
∞ ∞
−∞ ∞
f (x, y) dy dx = 1 f (x, y) ≥ 0
pmf
f (x, y) = 1 f (x, y) ≥ 0
x,y
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11. S = {(x, y) : f (x, y) 0}
The support or sample
space
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12. P (a X b, c Y d) =
d b
f (x, y) dx dy
c a
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13. What is the cdf
going to look like?
P (X x, Y y) =
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14. What is the cdf
going to look like?
P (X x, Y y) =
x y
F (x, y) = f (u, v)dvdu
−∞ −∞
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16. Important bits
Partial derivatives
Multiple integrals
(2d change of variable -
after spring break)
Use wolfram alpha. Wikipedia articles are
decent.
Friday, 26 February 2010
17. Your turn
F(x, y) = c(x 2 + y 2) -1 x, y 1
What is c?
What is f(x, y)?
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18. Marginal distributions
fX (x) = f (x, y)dy
R
fY (y) = f (x, y)dx
R
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19. Independence
How can we tell if two random variables
are independent?
Need to go back to our definition.
Friday, 26 February 2010
20. Dependence
Only one way for rv’s to be independent.
Many ways to be dependent. Useful to
have some measurements to summarise
common forms of dependence.
Next time we’ll use one you’ve hopefully
heard of before: correlation, a
measurement of linear dependence.
Friday, 26 February 2010
