Question

Wages (W) and years of education (X) have the following joint distribution: The fractions in the table are probabilities. That is, each cell is the probability that the row and column outcome happen at the same time. • (4 points) Find the marginal probability distribution for wages W. Hint: W takes on four values, you need to find the probability that it equals each one. • (4 points) Find the marginal probability distribution function for education X. • (5 points) Find the conditional expectation E(W|X = x) for each value of X. It should be a single number for each value of x. • (4 points) Find E(W) from the marginal probability function you found earlier. • (5 points) Now suppose I told you about something called the ”law of iterated expectations”. It means the following is true: E(W) = E(E(W|X)) (1) Verify that this is the case by taking the expectation of the variable you generated in part c and comparing it to your answer from part d.

3. Wages (W) and years of education (X) have the following joint distribution: = 14 = 16 X = 18 ။ | X = 10 / = 12 I = 5 IT = 10 IT = 15 ) I = 20 ) ။ အခြား 100 ခြံ 5-8- 100 The fractions in the table are probabilities. That is, each cell is the probability that the row and column outcome happen at the same time. • (4 points) Find the marginal probability distribution for wages W. Hint: W takes on four values, you need to find the probability that it equals each one. • (4 points) Find the marginal probability distribution function for education X. • (5 points) Find the conditional expectation E(W X = 2) for each value of X. It should be a single number for each value of x. • (4 points) Find E(W) from the marginal probability function you found earlier. • (5 points) Now suppose I told you about something called the "law of iterated expecta- tions". It means the following is true: E() = EE(WIX)) TTA Verify that this is the case by taking the expectation of the variable you generated in part c and comparing it to your answer from part d.

Answer 1

Page No. Date Ans Joint Prob" dirth Toble IX -16 1x=18 P(x=x) 21/100 11100 31100 29/100 W-51101100 51100 W.10 6100 81100 w 3/100 slice W-20 1'ico 21100 TP(W=W) 20/100 20100 31100 17/100 71100 3/100 20/100 21100 51100 81100 51100 20/100 29/100 61100 lolico 20/100 zilico 100 = 1 (1) Marginal Proh" dirth for wages w. T P(W=.wo) = { f (x, w) where frew) joint prob" of two Liepw=s): € +(2,5) = 4(10,5) + f(12,5) + f(14,5) +f676,5) + F(18.5) 5 - You to time to to= 211100 Table i plwas) = 21/100 similarly s prw=lo) = 29/100 PcW715)=297100, P6.820/= 362 Marginal Probability distribution of woge; W. Th 5 0 is zo Talall IP(W=6) 2/100/29/100 29/100 21/100 1 1 (2 Marginal Prob" dist for education x. .: PCX=x) = f(x, w) Wheme f(x,w) joint prob°of x&W - P(x = 10) = f(10,W) = 4(10,5) + f(10,10) + f (00, 15/offert L a torre et bottes moto is P(x=10) =

Page No. Similarly PCX=121320 L P(x:1) = 20, PCxE16) = 20 = P(x=19) 52 Table Mirginel Bob" dist of education of 20 loc 13 Conditional Expection E (W/x=x) = Z w Ele Cuore) Now find condition List" Fudbe, (wla) = few ed en Afw.k? P(x=x) peysat fron) = 160, xe) = joint prob" list of w&x ☺ find bo P(x= x) = minginal proba disan of x. fwtaziofwix) = 1 /(201 W =5 féll (200 wzje loo 100 Fiedt (2) مر - برا I Gool! (20.) W=20 W = 20 fuly 20 10 = 4 26 = 36 20 = 3/6 zo izo for was for walo for wars for 0:20 3 - 20

Page No. Date L E (W/x = 10 ) = 2 w lolxsio (ope) - Gyú) + (1x ) + (15 X 2) +(7044) 3.2.5+3+ +1 = 2.5+ 3 +2.25 + 1 = 8.75 E(helx=10) = 8.75 9] UO o find full of w/x) = {(812) - 167280) - /11 (20) Ha () - for was for walo for Wall for w=20 2016 100 o 2o E ( Wlx = 12) = { w ful12 (wa) = (5x) + (108 m) + fisk.) +(2015) = 3 4 + & + 12 + 2 = 1.25+4 + 3.75 +2 ElW/X=12) 0

Page No. Date Similarly CD E (Wien)- E w fulxong (w/3) /+42073.) = (5x3d +10xt+fis y = 2 2 + 1 2 + 2 / +3 =0.75 +3.5 +5.25 +3 LE(Wx=14) = 12.5 2 3 . w W .. E (W1 x=16) = { foleto (win) - = (3*3.) +fox Hex_)+ femme - 4 + 5 + 2 1 1 2 3 4 5 = 0.5 +2.5 + 6 +5 I E (W1x=16) = 14 ' ) +2015) Efwlx=iR) = { w frollere false) = (5x7d) + (10x) + (*** = ŕ + Ž + 1 + 10 = I E (W/x=18) = 0.25 +1.5+ 4.5+ 10 16.25 = 16 Etwt offwut on top twau = (5x bd) + (10x * ) + (is ) +(20 **) 29 Leo = 1.05 + 2190 +4.35 +4.29 Erw) = 12.50 513]