EconRandomVariables2

EconRandomVariables2 - Introductory Statistics Stats 210...

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1 Introductory Statistics Stats 210 Random Variables II
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2 Relationships between TWO random variables So far we were interested in ONE random variable Now we want to estimate the relationship between TWO random variables Example: Relationship between wages and education Relationship between crime and policing The Joint Probability Distribution makes this possible
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3 Joint Probability Distribution Joint probability distribution of RVs = gives the probability that two RVs take on specific values simultaneously. Example Rolling a die two times Probability of 6 in first throw AND second throw f(X=6, Y=6)
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4 Ebert and Roper Ebert and Roper rate Movies Ratings: 3 (Pro), 2 (Mixed), 1 (Con) The joint probability distribution gives the probability that, for example, both rate the movie favorably Let’s look at the raw data
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5 Ebert Roper 3 2 1 Total 36 491 08 3 2 11 13 8 32 382 44 5 T o t a l 8 83 04 21 6 0 How do we calculate the probability that both give a “thumbs up” (3) to the movie? Divide occurrences (64) by the total number of votes (160) Pr(R=Pro, E=Pro)=64/160=.4 Repeat this for all cells to get the joint probability distribution
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6 Joint Probability Distribution Ebert Roper 3 2 1 Total 3 64/160= .40 9/160= .06 10/160= .06 .52 2 11/160= .07 13/160= .08 8/160= .05 .20 1 13/160= .08 8/160= .05 24/160= .15 .28 .55 .19 .26 1 Roper’s Marginal Probability Distribution Ebert’s Marginal Probability Distribution
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7 Marginal Probability Distribution The Marginal Probability Distribution is the counterpart to the probability distribution with one variable To obtain the marginal probability distribution one just sums over every event of the other variable’s distribution Pr(E=Pro) = Pr(P, P) + Pr(P,M) + Pr(P, C) Formally: = = = = = k i i x X y Y p y Y p 1 ) , ( ) (
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8 Marginal Probability Distribution Ebert Roper 3 2 1 Total 3 64/160= .40 9/160= .06 10/160= .06 .40 + .06 + .06 = .52 2 11/160= .07 13/160= .08 8/160= .05 .07 + .08 + .05 = .20 1 13/160= .08 8/160= .05 24/160= .15 .08 + .05 + .15 = .28 .55 .19 .26 1 Roper’s Marginal Probability Distribution
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9 Yes, another Probability Distribution The next probability distributions has a prominent place in economics: Conditional Probability Distribution What is the probability of X given a particular value of Y Gives the conditional probability for each realization of a random variable p(Ebert | Roper=1)
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10 Conditional Probability Distribution Formally ) ( ) , ( ) | ( x X p x X y Y p x X y Y p = = = = = = Joint Probability Distribution Marginal Probability Distribution of X Conditional Probability Distribution of X
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11 Conditional Probability: Example Back to Ebert and Roper
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EconRandomVariables2 - Introductory Statistics Stats 210...

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