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Hadley Wickham
Stat310Bivariate random variables
Friday, 26 February 2010
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
1. Introduction to bivariate random variables
2. The important bits of multivariate calculus
3. Independence
Friday, 26 February 2010
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
Friday, 26 February 2010
Friday, 26 February 2010
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?
Friday, 26 February 2010
f(x, y) = c a < x, y < b
How could we work out c?Is this a pdf?
Friday, 26 February 2010
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
f(x, y) ≥ 0
f(x, y) ≥ 0
pmf�
x,y
f(x, y) = 1
� ∞
−∞
� ∞
∞f(x, y) dy dx = 1
Friday, 26 February 2010
S = {(x, y) : f(x, y) > 0}The support or sample space
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P (a < X < b, c < Y < d) =� d
c
� b
af(x, y) dx dy
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What is the cdf going to look like?
P (X < x, Y < y) =
Friday, 26 February 2010
What is the cdf going to look like?
F (x, y) =� x
−∞
� y
−∞f(u, v)dvdu
P (X < x, Y < y) =
Friday, 26 February 2010
Multivariate calculus
Friday, 26 February 2010
Important bitsPartial derivatives
Multiple integrals
(2d change of variable - after spring break)
Use wolfram alpha. Wikipedia articles are decent.
Friday, 26 February 2010
Your turn
F(x, y) = c(x2 + y2) -1 < x, y < 1
What is c?
What is f(x, y)?
Friday, 26 February 2010
fX(x) =�
Rf(x, y)dy
fY (y) =�
Rf(x, y)dx
Marginal distributions
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Independence
How can we tell if two random variables are independent?
Need to go back to our definition.
Friday, 26 February 2010
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
Read 3.3 and 3.3.1
Friday, 26 February 2010