bivariate populations

Lecture 5 1

Econ 140Econ 140

Bivariate Populations

Lecture 5

Lecture 5 2

Econ 140Econ 140Today’s Plan

• Bivariate populations and conditional probabilities

• Joint and marginal probabilities

• Bayes Theorem

Lecture 5 3

Econ 140Econ 140A Simple E.C.P Example

• Introduce Bivariate probability with an example of empirical classical probability (ecp).

• Consider a fictitious computer company. We might ask the following questions:– What is the probability that consumers will actually buy

a new computer?– What is the probability that consumers are planning to

buy a new computer?– What is the probability that consumers are planning to

buy and actually will buy a new computer?– Given that a consumer is planning to buy, what is the

probability of a purchase?

Lecture 5 4

Econ 140Econ 140

certainnull

A Simple E.C.P Example(2)

• Think of probability as relating to the outcome of a random event (recap)

• All probabilities fall between 0 and 1:

1)(0 AP

• Probability of any event A is:

naaaaAnm

AP ...,, with )( 321

Where m is the number of events A and n is the number of possible events

Lecture 5 5

Econ 140Econ 140A Simple E.C.P Example(3)

• The cumulative frequency is: 1)( iaP

• The sample space (of a 1000 obs) looks like this:Actually PurchaseYes (b1) No (b2) Total

Plan Yes (a1) 200 50 250to Purchase No (a2) 100 650 750

Total 300 700 1000

• Before we move on we’ll look at some simple definitions

Lecture 5 6

Econ 140Econ 140A Simple E.C.P Example(4)• If we have an event A there will be a compliment to A which we’ll call A’ or B • We’ll start computing marginal probabilities

– Event A consists of two outcomes, a1 and a2:

21,aaA – The compliment B consists also of two outcomes, b1 and b2:

21,bbB – two events are mutually exclusive if both events cannot occur– A set of events is collectively exhaustive if one of the events must occur

Lecture 5 7


• Computing marginal probabilities

kBAPBAPBAPA ...)Pr( 21

Where k is some arbitrary large number

• If A = planned to purchase and B=actually purchased:P(planned to buy) = P(planned & did) + P(planned & did not)=

25.01000250

100050

1000200

Actually PurchaseYes (b1) No (b2) Total


Total 300 700 1000

Lecture 5 8


• If the two events, A and B, are mutually exclusive, then

)()()( BPAPAorBP – General rule written as:

– Example: Probability that you draw a heart or spade from a deck of cards

• They’re mutually exclusive eventsP(Heart or Spade) = P(Heart) + P(Spade) – P(Heart + Spade)=

50.021

5226

05213

5213

)()()()( BAPBPAPAorBP

Lecture 5 9


• Probability that someone planned to buy or actually did buy: use the general addition rule:

)()()()( BAPBPAPAorBP • If A is planning to purchase, and B is actually purchasing, we can plug

in the marginal probabilities to find

35.01000350

1000200

1000300

1000250

Actually PurchaseYes (b1) No (b2) Total


Total 300 700 1000

Joint Probability: P(A and B): Planned and Actually Purchased

Lecture 5 10

Econ 140Econ 140Conditional Probabilities

• Lets leave the example for a while and consider conditional probabilities.

• Conditional probabilities are represented as P(Y|X)

• This looks similar to the conditional mean function:

Xn

Y

• We’ll use this to lead into regression line inference, and then we’ll look at Bayes theorem

Lecture 5 11

Econ 140Econ 140Conditional Probabilities (2)

• Probabilities will be defined as

Kk

Jj

kYYjXXPjkp

,...1

,...1

),(

• If we sum over j and k, we will get 1, or:

j k jkp 1

• We define the conditional probability as f (X|Y)– This is read “a function of X given Y”– We can define this as:

YXf |Y ofy probabilit MarginalY& X ofy probabilitJoint

Lecture 5 12


• Similarly we can define f (Y|X):

• Looking at our example spreadsheet, we have a sample of weekly earnings and years of education: L5_1.XLS.

• There are two statements on the spreadsheet that will clarify the difference between a joint and conditional probabilities

XYf |X ofy probabilit MarginalX& Y ofy probabilitJoint

Lecture 5 13


• The joint probability is a relative frequency and it asks: – How many people earn between $600 and $799 and have 10 years of education?

• The conditional probability asks: – How many people earn between $600 and $799 given they have 10 years of education?

• On the spreadsheet I’ve outlined the cells that contain the highest probability in each completed years of education– There’s a pattern you should notice

Lecture 5 14


• We can use the same data to graph the conditional mean function– the graph shows the same pattern we saw in the outlined cells– The conditional probability table gives us a small distribution

around each year of education

Lecture 5 15


• To summarize, conditional probabilities can be written as

)|(X ofy probabilit Marginal

Y& X ofy probabilitJoint )(

)&(YXf

XPYXP

– This is read as “The probability of X given Y”– For example: The probability that someone earns between $200 and $300, given that he/she has completed 10 years of

education• Joint probabilities are written as P(X&Y)

– This is read as “the probability of X and Y”– For example: The probability that someone earns between $200 and $300 and has 10 years of education

Lecture 5 16

Econ 140Econ 140A Marketing Example

• Now we’ll look at joint probabilities again using the marketing example from earlier in the lecture.

• We will look at:– Marginal probabilities P(A) or P(B)– Joint probabilities P(A&B)– Conditional probabilities

)()&(

BPBAP

Lecture 5 17

Econ 140Econ 140Marketing Example(2)

• Here’s the matrix

Actually PurchaseYes No Total

Plan Yes 200 50 250to Purchase No 100 650 750

Total 300 700 1000

• Let’s look at the probability you purchased a computer given that you planned to purchase:

%808.250200purchase) toplanned | purchased P(actually•

• The joint probability that you purchased and planned to purchase: 200/1000 = .2 = 20%

Lecture 5 18

Econ 140Econ 140Marketing Example (3)

• We can also represent this in a decision tree

Actually

Purchased

Actually

Purchased

P(AandB)= 200/1000

P(A’andB)= 100/1000

P(A’andB’)= 650/1000

P(AandB’)= 50/1000

Marginal Probabilities

Joint Probabilities

B

A

B

Lecture 5 19

Econ 140Econ 140Statistical Independence

• Two events exhibit statistical independence if

P(A|B) = P(A)

• We can change our marketing matrix to create a situation of statistical independence:

Actually PurchaseYes No Total

Plan Yes 75 175 250to Purchase No 225 525 750

Total 300 700 1000

Note: all we did was change the joint probabilities

30.0025.

075.

1000250

100075

)|( BAP 30.1000

300)( AP

30.0)()|( APBAP

Lecture 5 20

Econ 140Econ 140Sampling w/ and w/o Replacement

• How would sampling with and without replacement change our probabilities?

• If we have 20 markers (14 blue and 6 red)

– What’s the probability that we pick a red pen?

P(BR)=6/20

– If we replace the pen after every draw, what’s the probability that we pick red twice in a row?

(6/20)(6/20)=36/400 = .09 = 9%

– What’s the probability of drawing two reds in a row if we don’t replace after each draw?

(6/20)(5/19) = 30/380 =.079 = 7.9%

Lecture 5 21

Econ 140Econ 140Bayes Theorem

• With decision trees we had to know the probabilities of each event beforehand

• Using Bayes we can update using complement probabilities

• Consider the multiplication of independent events:

)()|(

)()()&(

APBAP

BPAPBAP

• The marginal probability rule says:)(...)2()1()( kBAPBAPBAPAP

Lecture 5 22

Econ 140Econ 140Bayes Theorem (2)

• Because of independence we can write P(A) another way)()|()2()2|()1()1|()( kBPkBAPBPBAPBPBAPAP

• We can now write our conditional probability function as:

)(

)()|()|(

AP

BPBAPBAP

)()|()()|()()|()()|(

2211 kk BPBAPBPBAPBPBAPBPBAP

• Plugging in our expression for P(A) gives us Bayes Theorem:

Lecture 5 23


• Think of the Bayes Theorem as probability in reverse

– You can update your probabilities in light of new information

• Suppose you have a product with a known probability of success

P(success) = P(S) = 0.4

P(failure) = P(S’) = 0.6

• We also know that a consumer group will write either a favorable or unfavorable report on the product

P(F|S) = 0.8 P(F|S’) = 0.3

Lecture 5 24


• Given our information, we want to find the probability that the product will be successful given a favorable report

P(S|F)

• In this case, Bayes says

)'()'|()()|(

)()|()|(

SPSFPSPSFP

SPSFPFSP

• We can plug values into the above equation to find

%6464.50.

32.

)6.0)(3.0()4.0)(8.0(

)4.0)(8.0()|(

FSP

• We can use the theorem to update the probability of a successful product given that the product gets a favorable report

Lecture 5 25

Econ 140Econ 140Recap

• We’ve seen how we can calculate marginal, joint, and conditional probabilities

– Computer company example

– Spreadsheet: L5_1.XLS

• We talked about statistical independence

• We’ve seen how Bayes Theorem allows us to update our priors

bivariate populations

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