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Bivariate Populations

Bivariate Populations. Lecture 5. Today’s Plan. Bivariate populations and conditional probabilities Joint and marginal probabilities Bayes Theorem. A Simple E.C.P Example. Introduce Bivariate probability with an example of empirical classical probability (ecp).

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Bivariate Populations

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  1. Bivariate Populations Lecture 5

  2. Today’s 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: null certain • 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 we’ll 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 we’ll call A’ or B • We’ll 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 B=actually 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 • They’re 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(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. 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. 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. 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. 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, giventhat 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 $300and has 10 years of education

  16. 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. 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. Marketing Example (3) • We can also represent this in a decision tree

  19. 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. 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(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%

  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(F|S) = 0.8 P(F|S’) = 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(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. 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

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