Econ 299 Chapter3b - 3.6 Joint Probability Density...

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1 3.6 Joint Probability Density Functions Sometimes we are interested in the isolated  occurrence or effects of one variable.  In this  case, a simple pdf is appropriate. Often we are interested in more than one  variable or effect.  In this case it is useful to use: Joint Probability Density Functions Conditional Probability Density Functions
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2 3.6 Joint Probability Density Functions Joint Probability Density Function - -summarizes the probabilities associated  with the outcomes of pairs of random variables f(p,q) = Prob(P=p and Q=q) ∑ f(p,q) = 1 Similar statements are valid for continuous  random variables.
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3 3.6 Joint PDF and You Love and Econ Example: On Valentine’s Day, Jonny both wrote an Econ  299 midterm and sent a dozen roses to his love  interest.   He can either pass or fail the midterm, and his  beloved can either embrace or spurn him.  E = {Pass, Fail}; L = {Embrace, Spurn} 
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4 3.6 Joint PDF and You Love and Econ Example: Joint pdf’s are expressed as follows: P(pass and embrace) = 0.32 P(pass and spurn) = 0.08 P(fail and embrace) = 0.48 P(fail and spurn) = 0.12 Notice that: ∑f(E,L) = 0.32+0.08+0.48+0.12 = 1
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5 3.6 Joint and Marginal Pdf’s Marginal (individual) pdf’s can be determined  from joint pdf’s.  Simply add all of the joint  probabilities containing the desired outcome of  one of the variables. Ie: f(Y=7)=∑f(Y=7,Z=z i ) Probability of Y=7 = sum of ALL joint probabilities  where Y=7
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6 3.6 Love and Economics f(pass) = f(pass and embrace)+f(pass and spurn)     = 0.32 + 0.08     = 0.40 f(fail)  = f(fail and embrace)+f(fail and spurn)    = 0.48 + 0.12    = 0.60 Find f(embrace) and f(spurn)
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7 3.6 Love and Economics Notice: Since passing or failing are exhaustive outcomes,  Prob (pass or fail) = 1 Also, since they are mutually exclusive, Prob (pass or fail)  = Prob (pass) + Prob (fail) = 0.4 + 0.6 = 1
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8 3.6 Conditional Probability Density  Functions Conditional Probability Density Function - -summarizes the probabilities associated  with the possible outcomes of one random  variable conditional on the occurrence of a  specific value of another random variable Conditional pdf = joint pdf/marginal pdf Or (Probability of “a” GIVEN “b”)
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9 3.6 Conditional Love and Economics From our previous example: Prob(pass|embrace) = Prob(pass and embrace)/ Prob (embrace) = 0.32/0.80 = 0.4 Prob(fail|embrace) = Prob(fail and embrace)/ Prob (embrace) = 0.48/0.80 = 0.6
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10 3.6 Conditional Love and Economics From our previous example: Prob(spurn|pass) = Prob(pass and spurn)/ Prob (pass) = 0.08/0.40 = 0.2 Prob(spurn|fail) = Prob(fail and spurn)/ Prob (fail) = 0.12/0.60 = 0.2
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Econ 299 Chapter3b - 3.6 Joint Probability Density...

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