This document provides an overview of bivariate random variables and related concepts for a statistics course. It discusses that bivariate random variables involve measuring two related quantities in a random experiment. Key concepts covered include the bivariate probability density function (pdf), calculating probabilities, marginal distributions, independence of random variables, and dependence. Independence means the joint pdf factors into the product of the marginal pdfs. Dependence will be further explored next time using the concept of correlation as a measure of linear dependence.

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
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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

5. Friday, 26 February 2010

6. 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)
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8. f (x, y) = c a < x, y < b
Is this a pdf?
How could we work out c?
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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.
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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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15. Multivariate calculus
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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.
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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.
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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.
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21. Read 3.3 and 3.3.1
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