Question

2. Two chips are chosen randomly with replacement from an urn containing 2 red, 3 black chips, and 5 gray chips. Let X denote the number of red chips chosen and let Y denote the number of black chips chosen. a. Find the joint probability mass function of X and Y , p(x,y) = P(X = 2, Y = y). b. Use the joint PMF to find the probability mass function, px(2), of X. c. Use the joint PMF to find the probability mass function, py(y), of Y. d. Find P(X > 0, Y = 2). e. Find P(X<2, Y > 0). f. Find E(X).

Answer 1

2 = lo ES Red Black 3. Total Gray chips chosen number of red chips 14 - number of black chips 2 are randomly (۵) - 2 كرا O Yy У(о 15110061100 19/100 Total 397100 21/100 9/100 2 3 Total 49/00 | 16/100 4/100 6%100 (6) Px (ne) Marginal of x Px (2=o) ५१ 100 P x (x = 1) 16 100 Px. (a = a) = 4 100

(c) Puly), marginal of 4 py(0) 39 100 Pul) E 2) 하히의 Pu(2) 9 200 Pu (3) = (d) P( x 70 , 1 = ) (e) (x <a, 170) 70) - P(x=0, 4-1) + P(x=0, 4-2) + P(x=o, Y= 3) + P(x, y = 1) + P(x =), 4-3) + Pk=/4=33) IS 1000 十 오 10 + + + + 0 100 30 100 0 300

W (1) E(x) — R () - ох ч9 | ФГrk + 2-х. • | В 9) О» 4 40 | Too 2 9 40 E(S) - Ога у