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Hadley Wickham
Stat310Bivariate transformations
Thursday, 19 February 2009
1. Recap
2. Probability as area/volume
3. Conditional pdfs
4. Transformations
Thursday, 19 February 2009
How can you simplify E(X + Y)? When can you do that?
When can you simplify E(XY) ?
Recap
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f(x, y) = f(y|x) f(x)f(x|y) f(y)
If X and Y are independent:
f(x, y) = f(x) f(y)
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Example
Oscar has a bad gambling problem. Every night on the way home from work he takes the X hundred dollars he earned at work that day and goes to the local casino. Oscar never wins any money but eventually stops playing to return home with Y hundred dollars.
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f(x, y) =18
0 < y < x < 4
f(x) =x
80 < x < 4
f(y) =18(4! y) 0 < y < 4
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Question
One night Oscar returns home with less than $200. What is:
The probability he started with less than $200?P(X < 2 | Y < 2)
The probability he lost more than $100?P(Y - X < 1 | Y < 2)
The probability he lost exactly $75?P(Y - X = 0.75 | Y < 2)
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Probability as area
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Original volume Conditioning boundary
New sample space Rescale to volume 1Thursday, 19 February 2009
Transformations
• Are (X - Y)/X and X independent?
• Let A = (X - Y)/X and B = X
• Are A and B independent? How can we tell?
• Need to produce two new random variables from two existing random variables: transformation
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Transformations
Distribution function
technique
Change of variable
technique
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Why is the cdf less useful in 2d?
P(x1 < X < x2, y1 < Y < y2)
P(X2 + Y2 < 1)
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P(x1 < X < x2, y1 < Y < y2) =
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F(x2, y2)
P(x1 < X < x2, y1 < Y < y2) =
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F(x2, y2)
- F(x1, y2)
P(x1 < X < x2, y1 < Y < y2) =
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F(x2, y2)
- F(x1, y2)
- F(x2, y1)
P(x1 < X < x2, y1 < Y < y2) =
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F(x2, y2)
- F(x1, y2)
- F(x2, y1)
+ F(x1, y1)
P(x1 < X < x2, y1 < Y < y2) =
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Change of variable technique
A = u1(X, Y )B = u2(X, Y )
X = v1(A, B)Y = v2(A, B)
fA,B = fX,Y (v1(A, B), v2(A, B))|J |
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J =!!!!
!x!a
!x!b
!y!a
!y!b
!!!!
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Back to the problem
• Are (X - Y)/X and X independent?
• What are A and B?
• What are u1 and u2?
• What are v1 and v2?
• What is fA,B(a, b) ?
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In general
• Figuring out v1 and v2 and their partial derivatives is usually easy
• Figuring out the bounds of integration is often hard!
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Problem
f(x, y) = 2 0 < x < y < 1
A = X/Y B = X
What are the bounds of A and B?
What is f(a, b) ?
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