The following table gives the joint probability distribution between employment

Question

The following table gives the joint probability distribution between employment status and college graduation among those either employed or looking for work (unemployed) in the working age U.S. population. Non-college grads (X= 0) College grads (X-1) Total 0.0537 0.0130 0.0667 0.6186 0.3147 0.933 Total 0.6723 0.3277 0.9997 The expected value of Y, denoted E(V, is(Round your response to three decimal places.) The unemployment rate is the fraction of the labor force that is unemployed. Show that the unemployment rate is given by 1-E( Unemployment rates 1-L-1-E(Y) = 1-0.933-0.0667 E(Yl X. 1) is (Roundyour response to three decimal places.) E(WX-o) is .(Roundyour response to three decimal places.) The unemployment rate for college graduates isand the unemployment rate for non-college graduates is(Round your responses to three decimal places) A randomly selected member of this population reports being unemployed. The probability that this worker is a college graduate is and the probability that this worker is a non-college graduate is [ (Round your responses to three decimal places.)

Explanation / Answer

E(Y) = 0. P(Y=0) + 1. P(Y=1) = P(Y =1)

= 1. 0.933 = 0.933

Unemployment rate = Number of people unemployed / ( Employed + Unemployed) = P( Y=0)

= 1- P(Y = 1) = 1 - E(Y)

Conditional Expectation of Y given X

P(Y = 0 / X=1) = P(Y=0,X=1) / P(X=1) = 0.0130/0.3277 = 0.040

P(Y=1/X=1) = P(Y=1, X =1)/P(X=1) = 0.3147 / 0.3277 =0.960

E(Y/X = 1) = 1.P(Y=1/X=1) + 0. P(Y=0/X=1) = P(Y=1/X=1) = 0.960

Unemployedment rate for college graduate = 1 - E(Y/X = 1) = 0.040

Unemployedment rate for Non College Graduates

P(X=0) is sum of the first row. Marginal probablity. P(X=0 ) = 0.6723

P(Y = 0 / X=0) = P(Y=0,X=0) / P(X=0) = 0.0537/0.6723 = 0.080

P(Y=1/X=0) = P(Y=1, X =0)/P(X=0) = 0.6186/ 0.6723 =0.920

E(Y/X = 0) = 1.P(Y=0/X=0) + 0. P(Y=0/X=0) = P(Y=1/X=0) = 0.920

Unemployedment rate for non college graduate = 1 - E(Y/X = 0) = 0.080

Probability that an unemployed person is college graduate

P(X=1/Y=0) = P(Y=0/X=1).P(X=1)/ P(Y=0) ( Bayes theorem)

=0.040 * 0.3277 /0.0667 = 0.197

Probability that an unemployed person is non college graduate

P(X=0/Y=0) = P(Y=0/X=0).P(X=0)/ P(Y=0)

=0.080 * 0.6723 /0.0667 = 0.806

Alternatively P(X=0/Y=0) = 1- P(X=1/Y=0)

Conditional Dependence

We have seen from the above example that P(X=0/Y=0) is not equal to P(X=0). The same result can be establised for P(X=0/Y=1) also. The two events X and Y are not independent since P(X=0/Y=1) is not equal to P(X=0).

Hence C is the right answer

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