12 bivariate
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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
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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
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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?
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f(x, y) = c a < x, y < b
How could we work out c?Is this a pdf?
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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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f(x, y) ≥ 0
f(x, y) ≥ 0
pmf�
x,y
f(x, y) = 1
� ∞
−∞
� ∞
∞f(x, y) dy dx = 1
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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) =
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What is the cdf going to look like?
F (x, y) =� x
−∞
� y
−∞f(u, v)dvdu
P (X < x, Y < y) =
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Multivariate calculus
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Important bitsPartial derivatives
Multiple integrals
(2d change of variable - after spring break)
Use wolfram alpha. Wikipedia articles are decent.
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Your turn
F(x, y) = c(x2 + y2) -1 < x, y < 1
What is c?
What is f(x, y)?
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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.
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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
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