Question

A restaurant serves three fixed-price dinners costing $12, $14, and $20. For a randomly selected couple dining at this restaurant, let x - the cost of the man's dinner and Y-the cost of the woman's dinner. The joint pmf of X and Y is given in the following table. p(x,y) 20 12 14 20 12 0.10 0.00 0.10 14 0.05 0.10 0.05 0.35 0.20 0.05 (a) Compute the marginal pmf of X. 12 14 20 Compute the marginal pmf of Y. y 12 14 20 py) (b) What is the probability that the man's and the woman's dinner cost at most $14 each? (c) Are X and Y independent? Justify your answer. X and Y are not independent because P(x, y) + Dy(x) - Dyky). X and Y are independent because P(x, y) = y(x) Dy(V), X and Y are independent because P(x, y) - Py(x) Pyly). X and Y are not independent because Alx, y) - Dx(x) D (). (d) What is the expected total cost, in dollars, of the dinner for the two people? S expensive meal that you (e) Suppose that when a couple opens fortune cookies at the conclusion of the meal, they find the message 'You will receive as a refund the difference between the cost of the more expensive and the have chosen." How much would the restaurant expect ta refund, in dollars? S

Answer 1

Solution ។ 2.4.4) 24 Qo Px (1) 12. 0.10 0.05 0:35 0.5 x 0:10 0.20 14 0.00 0.30 0.10 0.05 0.os Qo 0.2 0.2 0 2 0.6 سلم Pycy) 14 (a) Х 0.2 0.5 0:30 P (1) 14. 20 Y 0.6 0.2 0.2 Pyly) (b) PC* *514 and 4514) P(x=120 4:19) + P(X210,4=14) + PC s = 14, Y-12) + P(x214, 4-14) ) 0.25 0.10 +0.05 +0.00+ 0.10

(c) P (x = 12. Y=12) - 0.10 o P (x=12). PC yala) = 0.5x0.2 0.10. .pa x = 14,4=12)=0 = 0.06 P(x=14). p(y=12) = 0.30 x 0.20 P(x = 14, Y=12) + P(x214). PCY=12). inceperders rot are X and ។ not independent pCX, 4)+ P(x) • Pycy). x and ។ (d) E(x+4): ecx) + Ecy) Žy Pyly) { x Px (x) + 4 2 12 x0.2 + 14 x0.2 +20x6.6 12x o.5+ 14*0.30 + 2O x 0.2+ 31.4