1. Econ 140 Lecture 51 Bivariate Populations Lecture 5

2. Econ 140 Lecture 52 Today’s Plan Bivariate populations and conditional probabilities Joint and marginal probabilities Bayes Theorem

3. Econ 140 Lecture 53 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. Econ 140 Lecture 54 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: Probability of any event A is: Where m is the number of events A and n is the number of possible events

5. Econ 140 Lecture 55 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 we’ll look at some simple definitions

6. Econ 140 Lecture 56 A 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, a 1 and a 2 : –The compliment B consists also of two outcomes, b 1 and b 2 : –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. Econ 140 Lecture 57 A Simple E.C.P Example(5) Computing marginal probabilities 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)=

8. Econ 140 Lecture 58 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 They’re mutually exclusive events P(Heart or Spade) = P(Heart) + P(Spade) – P(Heart + Spade)=

9. Econ 140 Lecture 59 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. Econ 140 Lecture 510 Conditional 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: We’ll use this to lead into regression line inference, and then we’ll look at Bayes theorem

11. Econ 140 Lecture 511 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 (X|Y) –This is read “a function of X given Y” –We can define this as:

12. Econ 140 Lecture 512 Conditional Probabilities (3) 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

13. Econ 140 Lecture 513 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 I’ve outlined the cells that contain the highest probability in each completed years of education –There’s a pattern you should notice

14. Econ 140 Lecture 514 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. Econ 140 Lecture 515 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(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

16. Econ 140 Lecture 516 A 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

17. Econ 140 Lecture 517 Marketing Example(2) Here’s the matrix Let’s 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. Econ 140 Lecture 518 Marketing Example (3) We can also represent this in a decision tree

19. Econ 140 Lecture 519 Statistical 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: Note: all we did was change the joint probabilities

20. Econ 140 Lecture 520 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) –What’s the probability that we pick a red pen? P(B R )=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%

21. Econ 140 Lecture 521 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. Econ 140 Lecture 522 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. Econ 140 Lecture 523 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(F|S) = 0.8 P(F|S’) = 0.3

24. Econ 140 Lecture 524 Bayes Theorem (4) 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 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. Econ 140 Lecture 525 Recap 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