Question

Problem 4: Of ten police officers at a precinct, four are married, three have never married, and three are divorced. Three of the officers are to be selected for promotion. Let Y1 denote the number of married officers and Y2 denote the number of never-married officers among the three selected for promotion. Assume that the three officers are randomly selected from the ten available (1) Find the joint probability function of Yi and Y2. (2) Find the marginal probability distribution of Yı, the number of married officers among the three selected for promotion. 1|Y2 = 2) (3) Find P(Yi (4) If we let Y3 denote the number of divorced officers among the three selected for pro- motion, then Y3 = 3 - Yi - Y2. Find P(Y3 1Y2 1) (5) Compare the marginal distribution derived in (b) with the hypergeometric distributions 10, n 3 and r = 4 (see chapter 3 lecture notes on hypergeometric with N distribution) (6) Are the random variables Y1 and Y2 independent? (7) Find the expected number of married officers among the three selected for promotion. (8) Find Cov(Yı, Y2) (9) Find E(Y1 + Y2) and V(Y1 + Y2) by first finding the distribution of Y1 Y2

Answer 1

solution Given Data 4, denote y denote the number the number of married officers and of never - married officers so total 10 officers in that u married and 3 never married and 3 dévorcedes - 3 officers are to be selected which can be done in 10 ways live number of points in the Same Space - 1063) Ply=0 142=0) =P (weither married nor never married officers are selected p (All are divorced) - = 3 c3 I = 101 C10-3):31 - 1x ! 109x8*7*6*5X48 1003 PC 4,= 1, 4,-0) - Pli married none never married) (1 married and 2 divorced are selected)

6-2)!2! Toca 1003 . [: Permeting To (n-) 10! (10-3)! 31 3x4 цк у 19 120 10 x8x8x7 2x8x2 120 1: (4, - one out of 4 married obbickys can be selected - in yc, ways) (a 3 = 2 out of a divorced officers can be selected in See ways) ] Proceeding in are this manner the joint Probabilities obtained as follows 4,14,

4,142 ® The married marginal Probability distribution of Y., the number of obbicens among the thuee selected 687 Promotion, Pl 4;=Y) 201120-46 60/120 = 1/2 36/126 = 350

4/120 = 430 T ☺ PC4, = 1|42=2) PC 4,-1, 42=2) PC42=2) [ folmula using by definition) 12 To 12 x 120 [P(4, =1/42=2) = 4/4 among 4 she the 4₂ denote the number of divorced obbicers there selected böt Promotion , 43= 3-4, -4₂ - PC 43=1142 =1) = P(43=1, 42 = 1) P (42=1) P(41=1, 42=1, Y=1) P(Y2 = 1)

: one divorced + one selected means that the married), never married otticey is memaining oldites selected if 14 x 3c, x 3 W nocz Pl 43 = 1 | 4₂ = 1) = - 63/100 s (from robove (from tabove table) ux 3x3 36/120 120 = = 63/120 63/120 - 36/3 = 44 FP (4=114-1) - 44 cov (41,42) fólmula is cov (x, y) = f(xkly) - E(X)E(Y) je cov (4,4)= €14, 4.)= E(4) € (42) € (4,4)= 4,50 92=0 44P(958, 97-Y) olled) + ol !zo) + 0(°no) +olyna

to (140) + 1195hed) + 2 ( ") + 0 ( "ho) +214%) e = 0+0+0+0+0+ 36/120 + 4 + iko to = 36+ 20+36 120 36424036 120 = 0.8 cov (4.,42) = € (4,42) - €(41) (414) = 0.8-1.2 (0.9) = -0.28 Go € (4itYz) y PC4,4%a7%d) = 0 (Viro) + (*no) + 2/690) + 3 (35he) 2 .2.1 1+ 21 + 138 + 105 120 120 € (4,4 %)* 15= 4.9 v (4,+ Yz) - € (4+4)*- €14,+4) 49- (21) = 4,9-4.41 -0.49