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 for September 2017 Unemployed (Y=0) 0.026 Employed (Y=1) 0.576 Non-college grads (X=0) College Grads (X=1) 0.009 0.389 a) Compute marginal probabilities of X and Y. b) Compute E(Y) and E(X) c) Calculate E(YIX=1) and E(YIX=0) d) A randomly selected member of this population reports being unemployed. What is the probability that this worker is a college graduate? A non-college graduate? e) Are educational achievement and employment status independent? Explain.

Answer 1

(a)

	Unemployed (Y=0)	Employed (Y=1)	Marginal Probabilities of X
Non- College grads (X=0)	0.026	0.576	0.602
College Grades (X=1)	0.009	0.389	0.398
Marginal Probabilities of Y	0.035	0.965	1

Marginal probabilities of X

When X is 0, P (X=0) = 0.602

When X is 1, P (X= 1) = 0.398

Marginal probabilities of Y

When Y is 0, P (Y=0) = 0.035

When Y is 1, P (Y= 1) = 0.965

(b) Compute the expected value of Y as follows.

E (Y) = [0 * P (Y=0) ] + [1 * P (Y =1) ]

E (Y) = [ 0 * 0.035] + [1 * 0.965]

E (Y) = 0.965

Compute the expected value of X as follows.

E (X) = [0 * P (X=0) ] + [1 * P (X =1) ]

E (X) = [ 0 * 0.602] + [1 * 0.398]

E (X) = 0.398

(c) Calculate the expected value of Y given that the X is 1.

E(Y|X = 1) = [0x P(Y = 0) X = 1)]+[1x P(Y = 1| X = 1)] = 0 +[1x P(Y = 1| X = 1)] = P(Y = 1| X = 1)

= P (Y = 1, X =1) / P(X=1)

= 0.389 / 0.398

= 0.977

Calculate the expected value of Y given that the X is 0.

E(Y| X = 0) = [0x P(Y = 0 | X = 0)]+[1× P(Y = 1| X = 0)] = 0 +[1x P(Y = 1| X = 0)] = P(Y = 10 X = 0)

= P (Y = 1, X =0) / P(X=0)

= 0.576 / 0.602

= 0.956

(d) Calculate the probability that an unemployed person is a college graduate.

P(X = 1 Y = 0) - P(X = 1, Y = 0) P(Y = 0)

= 0.009 / 0.035

= 0.257

Calculate the probability that an unemployed person is a non college graduate

P(X = 0 Y = 0) = P(X = 0, y = 0) P(Y = 0)

= 0.026 / 0.035

= 0.742

(e) Educational achievement and employment status are not independent because the unemployment rate would change as the educational achievement changes.