Hadley Wickham

Stat310Bivariate distributions

Friday, 26 February 2010

1. Test

2. Example

3. CDF

4. Expectation

5. Conditional distributions

6. Correlation

Friday, 26 February 2010

Test

Friday, 26 February 2010

Grade

n

0

5

10

15

10 15 20 25 30 35 40

ABC

Friday, 26 February 2010

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

Friday, 26 February 2010

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.

Friday, 26 February 2010

Example

Friday, 26 February 2010

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0 10 20 30 40 50

type● 3pt● driving layup● hook● jump● layup● turnaround jump

Friday, 26 February 2010

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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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0 10 20 30 40 50

hook

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0 10 20 30 40 50

Friday, 26 February 2010

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0 10 20 30 40 50

driving layup

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layup

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0 10 20 30 40 50

hook

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0 10 20 30 40 50

Friday, 26 February 2010

y

..count..

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Marginal distribution of y

Friday, 26 February 2010

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Friday, 26 February 2010

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10 20 30 40

Joint distribution of x and y2114 jumps

Friday, 26 February 2010

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10 20 30 40

Friday, 26 February 2010

x

y

5

10

15

20

25

30

10 20 30 40

Friday, 26 February 2010

x

y

5

10

15

20

25

30

made

10 20 30 40

missed

10 20 30 40

Friday, 26 February 2010

After spring break:

Will look at three pointers and useful transformations of the data

Friday, 26 February 2010

Conditional distributions

Friday, 26 February 2010

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

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

X | Y = y

Y | X = x

Friday, 26 February 2010

x

y

0

5

10

15

20

25

30

0 10 20 30 40 50

Friday, 26 February 2010

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

Friday, 26 February 2010

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

Friday, 26 February 2010

x

y

0

5

10

15

20

25

30

0 10 20 30 40 50

Friday, 26 February 2010

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

Friday, 26 February 2010

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

Friday, 26 February 2010

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) ?

Friday, 26 February 2010

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)

Friday, 26 February 2010

Joint

Marginal

Conditional

2d 1d

integ

ratio

n

fixing value

Friday, 26 February 2010

Why is the cdf less useful in 2d?

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

P(X2 + Y2 < 1)

Friday, 26 February 2010

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

(x1,y1)

(x2,y2)

Friday, 26 February 2010

F(x2, y2)

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

(x1,y1)

(x2,y2)

Friday, 26 February 2010

F(x2, y2)

- F(x1, y2)

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

(x1,y1)

(x2,y2)

Friday, 26 February 2010

F(x2, y2)

- F(x1, y2)

- F(x2, y1)

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

(x1,y1)

(x2,y2)

Friday, 26 February 2010

F(x2, y2)

- F(x1, y2)

- F(x2, y1)

+ F(x1, y1)

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

(x1,y1)

(x2,y2)

Friday, 26 February 2010

Expectation

Friday, 26 February 2010

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!

Friday, 26 February 2010

Correlation

Friday, 26 February 2010

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

Friday, 26 February 2010

Correlation = 0.8

Friday, 26 February 2010

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

Friday, 26 February 2010

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)?

Friday, 26 February 2010

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!)

Friday, 26 February 2010

Correlation

ρXY =σXY

σXσY

ρXY =σXY√

σXXσY Y

Friday, 26 February 2010

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?

Friday, 26 February 2010

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?

Friday, 26 February 2010