12 Bivariate

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


• 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


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


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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


! !

Sf(x, y) dy dx = 1

! !

S

f(x, y) = 1

f(x, y) ! 0

f(x, y) ! 0

pdf

pmf


fX(x) =!

Rf(x, y)dy

fY (y) =!

Rf(x, y)dx


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) =!

Rf(x, y)dy

fY (y) =!

Rf(x, y)dx

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 )) =! !

Su(x, y) f(x, y) dy dx

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

2:30-4:30pm

Oshman Engineering Design Kitchen


Health & Medicine

12 Bivariate