The approximately 100 million adult Americans (age 25 and over in 1985) were rou

Question

The approximately 100 million adult Americans (age 25 and over in 1985) were roughly classified by education X and age Y as follows Age Y Education X (25-35) (35-55) (55-100) (last completed school) 30 45 70 none 0 1,000,000 2,000,000 5,000,000 primary 1 3,000,000 6,000,000 10,000,000 secondary 2 18,000,000 21,000,000 15,000,000 college 3 7,000,000 8,000,000,4,000,000 What is the probability of getting a 30-year-old college graduate (X=3 and Y=30)? And what is the probability of getting each of the 12 possible combinations of education and age? That is, tabulate the joint probability distribution P(x,y). (b) Calculate p(x) and p(y) (c) Are X and Y independent? (d) Calculate mu_x and sigma_x.

Explanation / Answer

a) P(X=3 and Y = 30) = 0.07

b) MPMF of X is P(X=x)

Put x =0

P(X=0) = 0.01 + 0.02 +0.05 = 0.08

P(X=1) = 0.03 + 0.06 +0.10 = 0.19

P(X=2) =0.18 + 0.21 +0.15 = 0.54

P(X=3) = 0.07 + 0.08 + 0.04 =0.19

The marginal pmf of Y is

P(Y=30) = 0.01+0.3+0.18+0.07 = 0.29

P(Y=45) = 0.02+0.06+0.21+0.08 = 0.37

P(Y=70) =0.05+0.10+0.15+0.04 = 0.34

c) X and Y not independent since P(X=x,Y=y) not = P(X=x).P(Y=y), for all x, y

d)

E(X)= Mean of X = 0*0.08+1*0.19+2*0.54+3*0.19 = 1.84

E(X2)=0*0.08+12*0.19+22*0.54+32*0.19 = 4.06

Var(X) = E(X2) - [E(X)]2

         = 4.06 - 1.842 = 0.6744

SD(X) = sqrt(0.6744) = 0.8212

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