13 Bivariate

Hadley Wickham

Stat310Bivariate distributions

Friday, 26 February 2010

1. Test

2. Example

3. CDF

4. Expectation

5. Conditional distributions

6. Correlation


Test


Grade

n

0

5

10

15

10 15 20 25 30 35 40

ABC


Grade

n

0

5

10

15

10 15 20 25 30 35 40

ABC

But, you can expecta boost of about 10%from your homeworks


TestIt was a bit harder than I intended (Question 1 was supposed to be the easiest ☹). I don’t curve, but I will provide opportunities for you to improve your overall grade.

Extra credit this week worth 5 points - you may redo one question from the exam.

Model answers will be posted after spring break.


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0 10 20 30 40 50

driving layup

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layup

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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●●●

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●

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●●●●●●●●●●●●● ●●●●●●●●●●

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hook

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0 10 20 30 40 50

driving layup

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layup

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hook

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0 10 20 30 40 50


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..count..

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10 20 30 40


x

y

5

10

15

20

25

30

10 20 30 40


x

y

5

10

15

20

25

30

made

10 20 30 40

missed

10 20 30 40


After spring break:

Will look at three pointers and useful transformations of the data


Conditional distributions


f(x|y) =f(x, y)f(y)

f(y|x) =f(x, y)f(x)

X | Y = y

Y | X = x


x

y

0

5

10

15

20

25

30

0 10 20 30 40 50


x

..count.. 0

10

20

30

40

0

10

20

30

40

[0,5]

(15,20]

10 20 30 40

(5,10]

(20,25]

10 20 30 40

(10,15]

(25,30]

10 20 30 40

Conditionaldistribution of X given Y


x

..count.. 0

10

20

30

40

0

10

20

30

40

[0,5]

(15,20]

10 20 30 40

(5,10]

(20,25]

10 20 30 40

(10,15]

(25,30]

10 20 30 40


x

y

0

5

10

15

20

25

30

0 10 20 30 40 50


y

..count.. 0

10

20

30

40

50

60

0

10

20

30

40

50

60

[0,10]

(30,40]

5 10 15 20 25 30

(10,20]

(40,50]

5 10 15 20 25 30

(20,30]

5 10 15 20 25 30

Conditionaldistribution of Y given X


y

..count.. 0

10

20

30

40

50

60

0

10

20

30

40

50

60

[0,10]

(30,40]

5 10 15 20 25 30

(10,20]

(40,50]

5 10 15 20 25 30

(20,30]

5 10 15 20 25 30


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)


Joint

Marginal

Conditional

2d 1d

integ

ratio

n

fixing value


Why is the cdf less useful in 2d?

P(x1 < X < x2, y1 < Y < y2)

P(X2 + Y2 < 1)


P(x1 < X < x2, y1 < Y < y2) =

(x1,y1)

(x2,y2)


F(x2, y2)

P(x1 < X < x2, y1 < Y < y2) =

(x1,y1)

(x2,y2)


F(x2, y2)

- F(x1, y2)

P(x1 < X < x2, y1 < Y < y2) =

(x1,y1)

(x2,y2)


F(x2, y2)

- F(x1, y2)

- F(x2, y1)

P(x1 < X < x2, y1 < Y < y2) =

(x1,y1)

(x2,y2)


F(x2, y2)

- F(x1, y2)

- F(x2, y1)

+ F(x1, y1)

P(x1 < X < x2, y1 < Y < y2) =

(x1,y1)

(x2,y2)


Expectation


E(aX + bY) = aE(X) + bE(Y)

E(XY) = E(X)E(Y) if X and Y are independent

expectation is still a linear operator!


Correlation


From: http://en.wikipedia.org/wiki/File:Correlation_examples.png


http://en.wikipedia.org/wiki/File:Correlation_examples.png

http://en.wikipedia.org/wiki/File:Correlation_examples.png

Correlation = 0.8


Covariance

Easiest to define correlation in terms of another function: covariance

Cov(X, Y) = E[ (X - E(X))(Y - E(Y)) ] = σXY

Cov(X, X) = Var(X) = σXX = σX2


Alternative

Is there another way to compute the covariance? (Think about the two ways of computing the variance)

If X and Y are independent, what is Cov(X, Y)?


Covariance

If σXY > 0, then X tends to increase when Y increases

If σXY < 0, then X tends to decrease when Y increases

If σXY = 0, then there is no linear relationship between X and Y (but there may be a non-linear relationship!)


Correlation

ρXY =σXY

σXσY

ρXY =σXY√

σXXσY Y


ρ

What is the range of ρ?

When are X and Y most strongly positively related?

When are X and Y most strongly negatively related?


Counterexample

Let X be a random variable with E(X) = 0, E(X2) = 10, E(X3) = 0. Let Y = X2.

Are X and Y independent?

Are X and Y uncorrelated?


Travel

13 Bivariate