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Hadley Wickham
Stat310Bivariate distributions
Thursday, 19 February 2009
1. Feedback
2. Test info
3. Example
1. Constructing a joint
2. Expectation
3. Independence
4. Conditioning
Thursday, 19 February 2009
• Still like the examples, and the recaps
• 10 said you found the help sessions really helpful
• 3 commented on my improved board skills :)
• 2 said speed was good, 1 too slow and 4 too fast
Feedback
Thursday, 19 February 2009
Feedback• Still don’t like me making mistakes.
• Would like more real-life and better prepared examples - will do.
• Website needs to be updated more frequently - I’ll try!
• Will try and put readings up ahead of time
Thursday, 19 February 2009
Casual Formal
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+ fant
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+ t-sh
irts:
still a
wesom
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- like
your
pre
vious
set o
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+ clot
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provin
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Thursday, 19 February 2009
Test info
Thursday 26th February
Same format as last test, but only one page of notes.
Continuous & bivariate random variables.
Will put more info on website asap.
Thursday, 19 February 2009
Recap
Thursday, 19 February 2009
! !
Sf(x, y) dy dx = 1
! !
S
f(x, y) = 1
f(x, y) ! 0
f(x, y) ! 0
pmf
Thursday, 19 February 2009
fX(x) =!
Rf(x, y)dy
fY (y) =!
Rf(x, y)dx
Thursday, 19 February 2009
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.
Thursday, 19 February 2009
Question
If X is a random variable with pdf f(x) = x / 8 0 < x < 4
and Y|X=x ~ Unif(0, x)
What is the joint pdf f(x, y) ?
Hint:P (X ! A " Y ! B) = P (X ! A|Y ! B)P (Y ! B)
Thursday, 19 February 2009
f(x, y) = f(y|x) f(x)f(x|y) f(y)
If x and y are independent, what does that imply about f(x, y) ?
Thursday, 19 February 2009
f(x, y) = f(y|x) f(x)f(x|y) f(y)
If x and y are independent, what does that imply about f(x, y) ?
f(x, y) = f(x) f(y)
Thursday, 19 February 2009
Question
What is f(y)? (how much money Oscar brings home)
i.e. imagine we don’t know X, but still want some idea of the likely amounts of money Oscar will bring home
Thursday, 19 February 2009
fX(x) =!
Rf(x, y)dy
fY (y) =!
Rf(x, y)dx
Be careful with limits of integration!
Thursday, 19 February 2009
S = {(x, y) : f(x, y) > 0}
Sx = {x : f(x) > 0}Sy = {y : f(y) > 0}
Thursday, 19 February 2009
Question
What is Oscar’s expected loss?
We don’t have the tools to solve this yet, but you can still convert the word problem to a mathematical problem.
And you can use your intuition to think about what would make sense
Thursday, 19 February 2009
E(u(X, Y )) =! !
Su(x, y) f(x, y) dy dx
So what is E(X - Y)?
Does that number make sense?
Thursday, 19 February 2009
Theory question
If X and Y are independent, what is E(XY) ?
What does that imply about the intuition we used in the previous problem?
Thursday, 19 February 2009
Theory result
E(XY) = E(X) E(Y)
So maybe that implies X and Z = (X-Y)/X (percent loss) are independent.
We’ll look at the tools to show that next time.
Thursday, 19 February 2009
Question
One night Oscar returns home with less than $200. What is:
The probability he started with less than $200?
The probability he lost more than $100?
The probability he lost exactly $75?
If you don’t see how to do this immediately, you can still write it out mathematically and use your intuition.
Thursday, 19 February 2009
P(X < 2 | Y < 2)
P(Y - X < 1 | Y < 2)
P(Y - X = 0.75 | Y < 2)
Thursday, 19 February 2009
Engineering Majors Day
2:30-4:30pm
Oshman Engineering Design Kitchen
Thursday, 19 February 2009
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