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Lottery:  A tax on people who are bad at math.  ~Author Unknown. Lecture 14: Multivariate Distributions. Probability Theory and Applications Fall 2008 October 17-20. Outline. Multivariate Distributions Bivariate Distributions Discrete Continuous Mixed Marginal Distributions

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Lecture 14 multivariate distributions

Lottery:  A tax on people who are bad at math.  ~Author Unknown

Lecture 14: Multivariate Distributions

Probability Theory and Applications

Fall 2008

October 17-20


Outline
Outline Unknown

  • Multivariate Distributions

  • Bivariate Distributions

    • Discrete

    • Continuous

    • Mixed

  • Marginal Distributions

  • Conditional Distributions

  • Independence


Multivariate distributions
Multivariate Distributions Unknown

Distributions may have more than one R.V.

Example: S=size of house - real RV

P=price of house - real RV

A=Age of house - real RV

C= condition of house

Excellent, Very Good, Good, Poor - discrete RV

Since variables are not-independent need a multivariate distribution to describe them: f(S,P,A,C)


Bivariate random variables
Bivariate Random Variables Unknown

Given R.V. X and Y

Cases

  • X,Y both discrete

    number of blue and red jelly beans

    picked from jar

    2. X,Y both continuous

    height and weight

    3. X discrete and Y continuous

    date and stock price


Both discrete
Both Discrete Unknown

The joint distribution of (X,Y) is specified by

  • The value set of (X,Y)

  • The joint probability function

    f(x,y)=P(X=x,Y=y)

    Note:

  • f(x,y)≥0 for any (x,y)


Discrete example
Discrete Example Unknown

3 H

2 M

2 D

Box contains jewels H=high quality

M=medium quality

D=defective

You pick two jewels w/o replacement

X=# of H

Y =#of M


Joint probability function
Joint Unknown Probability Function


Joint probability function1
Joint Unknown Probability Function


Marginal probability functions
Marginal Unknown Probability Functions


Definitions
Definitions Unknown

The marginal distribution of X is

Note this is exactly the same as pdf of X

The joint cumulative density function of X,Y is


Questions
Questions Unknown

P(You get one high quality and one medium jewel)?

P(You pick at least one high quality jewel)?


Conditional distributions
Conditional Distributions Unknown

The conditional distribution of Y given X is

In our example:


Conditional probability functions
Conditional Unknown Probability Functions


Conditional probability functions1
Conditional Unknown Probability Functions

Find distribution of

X given Y=1


Question
Question Unknown

Given that exactly one jewel picked is medium quality, what is the probability that the other is high quality?

6/10

Given that at least one jewel picked is medium quality, what is the probability that the other is high quality?

6/11


X y both continuous
X,Y Unknown both Continuous

The joint pdf, f(x,y) defined over R2has properties:

  • f(x,y)≥0

    To calculate probabilities, integrate joint pdf over X,Y over the area

    Or more generally if we want


X y both continuous1
X,Y Unknown both Continuous

More generally if we want

The c.d.f.


Marginals and conditionals
Marginals and Conditionals Unknown

The marginal pdf of X

The marginal pdf of Y

The conditional pdf of X given Y=y


Examples
Examples Unknown

The joint pdf of (x,y) is

Find c


Continued
continued Unknown

Find pdf of X

Find pdf of Y


Continued1
continued Unknown

Find marginal of X given Y=1

Note this is the same as marginal of X!

X and Y are independent!


Continued2
continued Unknown

2

Y

Find P(X>Y)

0 X 1


Mixed continuous and discrete
Mixed Continuous and Discrete Unknown

Let L a be R.V. that is 1 if candy corn manufactured from Line 1 and 0 if line 0

Let X=weight of candy corn

The joint pdf is

What is the marginal distribution of X – the weight of the candy corn?


Mixed continuous and discrete1
Mixed Continuous and Discrete Unknown

The joint pdf is

Sum over L to find the marginal of X


Conditional distribution
Conditional Distribution Unknown

What is the marginal of L?

L is Bernoulli R.V. p=0.25

What is the conditional X given L?

If candy corn is from Line 1,

weight is normal with

mean 7.05 and s.d. = 1.

If candy corn is from Line 0,

weight is normal with

mean 10.1 and s.d. = 1.2.


Mixture model
Mixture Model Unknown

X is a mixture of two different normals


Example 5
Example 5 Unknown

Harry Potter plays flips a magical coin 10 times and records the number of heads.

The coin is magical because each day the probability of getting heads changes.

Let Y, the probability of getting heads on a given day, be uniform [0,1]

Let X be the number of heads of 10 gotten on a given day with the magic coin.

What is the pdf of X?


Example 5 continued
Example 5 continued Unknown

Y is uniform [0,1] so

X|Y is binomial n=10 p=Y

So f(X,Y)

X is discrete uniform

All values equally likely


Fact Unknown

You can compute the joint from a marginal and a conditional.

Be careful how you compute the value sets!


Example 2 two continuous
Example 2 – Two Continuous Unknown

The joint pdf of X and Y is

Find marginal of X

1

Y

O

X

1


Example 2
Example 2 Unknown

Still need c

You check:


Continued3

1 Unknown

1

Y

Y

O

O

1

1

X

X

continued

P(Y≥2X)

Find P(Y<2X)


Conditional distribution1
Conditional distribution Unknown

Find conditional pdf of Y and X=1/2

1

Y

O

X

1


Conditional distribution2
Conditional distribution Unknown

Find conditional pdf of Y and X=x 0<x<1

1

Y

O

X

1


Independence
Independence Unknown

R.V. X and Y are independent if and only any of the following hold

  • F(x,y)=FX(x)FY(y)

    P(X≤x,Y≤y)= P(X≤x)P(Y≤y)

    2. f(x,y)=fX(x)fY(y)

    3. f(y|x)=fY(y)


Example 3
Example 3 Unknown

Given the joint pdf of X,Y

Use the marginal of X and the conditional pdf of Y given X=x to determine if X and Y are independent?


Answer
Answer Unknown

1

Find marginal of X

Find conditional of Y given X

Y

O


Answer continued
Answer continued Unknown

Are they independent?

No


Note Unknown

P(Y≤3/4|x=1/2) and P(Y≤3/4|x ≤1/2) are very different things!

Let’s calculate each one


P y 3 4 x 1 2
P(Y≤3/4|X=1/2) Unknown

The pdf of Y given X=1/2 is

so


P y 3 4 x 1 21
P(Y≤3/4|X ≤ 1/2) Unknown

The probability Y given X ≤ 1/2 is

where


P y 3 4 x 1 22
P(Y≤3/4|X ≤ 1/2) Unknown

The probability P(Y≤3/4,X ≤ 1/2)

The probability

1

O


Example 4
Example 4 Unknown

Suppose X has the Gamma distribution with parameters with K=2 and theta=1 and

the conditional distribution of Y given X.

(X>0) is

Find P( X<4| Y=2)


Example 41
Example 4 Unknown

We know f(x,y)=f(x|y)fx(x) so the joint is

The marginal of Y is

Thus conditional of X given Y is


Example 4 continued
Example 4 continued Unknown

So

Thus

Exercise try: P(X>4|Y>2)


Example 5 two discretes
Example 5 – Two Discretes Unknown

You write a paper with an average rate of 10 errors per paper. Assume the number of errors per papers follows a Poisson distribution.

You roommate proofreads it for you, and he/she has .8 percent of correcting each error.

What is the joint distributions of the number of errors and the number of corrections?

What is the distribution of the number of errors after you roommate reads the paper?


Answer1
answer Unknown

Let X be the number of errors

Y be the number of errors after correction

Clearly Y depends on X.

Given

What is pdf of Y|X?

binomial(n=X,p=.2)


Answer2
answer Unknown

Let X be the number of errors

Y be the number of errors after correction

Extra Credit: if you can figure out marginal of Y.


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