Question

$3.12 #2 Let D, i = 1,2,... denote the number of items of a certain kind sold on days 1, 2, dots, with the DE {1,2,3). Suppose data show that if a day's demand is 1 or 2, the next day's sales will be 1, 2 or 3 with probability 1 / 3 each. But on high-demand days, i.e. those on which 3 items are sold, the number sold the next day will be 1, 2 or 3, with probability 0.2, 0.2 and 0.6, respectively; in other words, the high demand has "momentum," with one 3-item day more likely to be followed by another. Say today 3 items were sold. What is the expected number of items to be sold two days from now? $3.12 #6 In the card game bridge, all 52 cards are dealt out evenly to four players. Find the expected number of players having no aces. $4.7 #4 Suppose X and Y are indicator random variables, i.e. modeling Bernoulli trials, with P(X = 1) = PſY = 1) = v, and such that P(X = Y = 1) = w. Find var(X+Y) in terms of u and w.

Answer 1

Scenario	Instances	Number of players with zero aces
One player has 4 aces, rest 0	4	3	0.11
Each player has one ace	1	0	0.03
2 players have 2 cards each, rest have 0	6	2	0.17
1 player has 3, 1 player has 1 and others 0	12	2	0.34
1 player has 2, 2 players have 1 each, one has 0	12	1	0.34
Expected number of players with zero cards	=3*0.11 + 0*.03 + 2*.17 + 2*.34 + 1*.34
	1.7

Hence 1.7 players are expected to zero aces