Bivariate Populations

1
Bivariate Populations

• Lecture 5

2
Todays Plan

• Bivariate populations and conditional
  probabilities
• Joint and marginal probabilities
• Bayes Theorem

3
A 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?

4
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

certain
null

• Probability of any event A is

Where m is the number of events A and n is the
number of possible events
5
A Simple E.C.P Example(3)

• The cumulative frequency is

• The sample space (of a 1000 obs) looks like this

• Before we move on well look at some simple
  definitions

6
A Simple E.C.P Example(4)

• If we have an event A there will be a compliment
  to A which well call A or B
• Well start computing marginal probabilities
• Event A consists of two outcomes, a1 and a2

• The compliment B consists also of two outcomes,
  b1 and b2

• two events are mutually exclusive if both events
  cannot occur
• A set of events is collectively exhaustive if one
  of the events must occur

7
A Simple E.C.P Example(5)

• Computing marginal probabilities

• Where k is some arbitrary large number

• If A planned to purchase and Bactually
  purchased
• P(planned to buy) P(planned did) P(planned
  did not)

8
A Simple E.C.P Example(6)

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

• General rule written as
• Example Probability that you draw a heart or
  spade from a deck of cards
• Theyre mutually exclusive events
• P(Heart or Spade) P(Heart) P(Spade) P(Heart
  Spade)

9
A Simple E.C.P Example(6)

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

• If A is planning to purchase, and B is actually
  purchasing, we can plug in the marginal
  probabilities to find

Joint Probability P(A and B) Planned and
Actually Purchased
10
Conditional Probabilities

• Lets leave the example for a while and consider
  conditional probabilities.
• Conditional probabilities are represented as
  P(YX)
• This looks similar to the conditional mean
  function

• Well use this to lead into regression line
  inference, and then well look at Bayes theorem

11
Conditional Probabilities (2)

• Probabilities will be defined as

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

• We define the conditional probability as f (XY)
• This is read a function of X given Y
• We can define this as

12
Conditional Probabilities (3)

• Similarly we can define f (YX)

• 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

13
Conditional Probabilities (4)

• 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 Ive outlined the cells that
  contain the highest probability in each completed
  years of education
• Theres a pattern you should notice

14
Conditional Probabilities (5)

• 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

15
Conditional Probabilities (6)

• To summarize, conditional probabilities can be
  written as

• 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(XY)
• 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

16
A Marketing Example

• Now well 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(AB)
• Conditional probabilities

17
Marketing Example(2)

• Heres the matrix

• Lets look at the probability you purchased a
  computer given that you planned to purchase

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

18
Marketing Example (3)

• We can also represent this in a decision tree

19
Statistical Independence

• Two events exhibit statistical independence if
• P(AB) P(A)
• We can change our marketing matrix to create a
  situation of statistical independence

Note all we did was change the joint
probabilities
20
Sampling 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)
• Whats the probability that we pick a red pen?
• P(BR)6/20
• If we replace the pen after every draw, whats
  the probability that we pick red twice in a row?
• (6/20)(6/20)36/400 .09 9
• Whats the probability of drawing two reds in a
  row if we dont replace after each draw?
• (6/20)(5/19) 30/380 .079 7.9

21
Bayes 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

• The marginal probability rule says

22
Bayes Theorem (2)

• Because of independence we can write P(A) another
  way

• We can now write our conditional probability
  function as

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

23
Bayes Theorem (3)

• 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(FS) 0.8 P(FS) 0.3

24
Bayes Theorem (4)

• Given our information, we want to find the
  probability that the product will be successful
  given a favorable report
• P(SF)
• In this case, Bayes says

• We can plug values into the above equation to
  find

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

25
Recap

• Weve seen how we can calculate marginal, joint,
  and conditional probabilities
• Computer company example
• Spreadsheet L5_1.XLS
• We talked about statistical independence
• Weve seen how Bayes Theorem allows us to update
  our priors