12 Bivariate

Stat310 Bivariate distributions

Hadley Wickham
Thursday, 19 February 2009

1. Feedback
2. Test info
3. Example
1. Constructing a joint
2. Expectation
3. Independence
4. Conditioning


Feedback
• 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
• Still don’t like me making mistakes.
• Would like more real-life and better
prepared examples - will do.
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frequently - I’ll try!
• Will try and put readings up ahead of
time


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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.


Recap


pdf

f (x, y) ≥ 0
f (x, y) dy dx = 1
S

pmf

f (x, y) ≥ 0
f (x, y) = 1
S


fX (x) = f (x, y)dy
R

fY (y) = f (x, y)dx
R


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.


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)


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 (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)


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


fX (x) = f (x, y)dy
R

fY (y) = f (x, y)dx
R

Be careful with limits of integration!

S = {(x, y) : f (x, y) > 0}

Sx = {x : f (x) > 0}
Sy = {y : f (y) > 0}


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


E(u(X, Y )) = u(x, y) f (x, y) dy dx
S

So what is E(X - Y)?
Does that number make sense?


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?


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.


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.


P(X < 2 | Y < 2)
P(Y - X < 1 | Y < 2)
P(Y - X = 0.75 | Y < 2)


Engineering Majors Day
Oshman Engineering Design Kitchen

2:30-4:30pm


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