Question

(4) A restaurant based in a city centre monitors the rate at which customers are arriving and leaving. It finds during a particular lunchtime that customers arrive at a mean rate of 4.5 in 30 minutes, while they leave at a mean rate of 2.2 in 30 minutes. Giving your answers to 2d.p. calculate the probability that a) exactly 6 customers arrive in one 30-minute period. b) at most 3 customers leave in one 30-minute period. c) between 2 and 6 customers exclusive leave in one 30-minute period. d) at most 12 customers arrive in a 1-hour period. e) between 5 and 7 customers inclusive leave in one and half hour period. [4]

Answer 1

Let .X and Y de note arriving and teaving time in 30 minutes of customen respectively. The arrive at me an rate of 4.5 in3o. BAS minutes orios oro so X of ~ poisson (4-5) sors rate of 2.2 The leave at a mean toin 30 minutes. so y ~ poisson (202) t o . The profof xis, P(x-x) = é 4. (4.5) * 2=0,1,2,- The pmf ofry is, Ply=y) = é a 4 (2+2) y=0,1,2, a exactly 6 custometers arrive in one 30 minute period. solution: Here, x ~ poisson (4.5) P(x=6) = &- 4.5 (4.5) 6 = 0.12812 ~ 0.13

6 at most. 3 customers leave in one 30 minutes periodi solution: Here y~ Poisson (2.2) P(Y€3) = P(4=0) + P(X=1) + PlY=2) + P(4:3) é 2. 2 2.2 + é x 2.2) é 2 02. 21 te 3) -2.27 1 + 2.2 + 2.2) + 2 e é 2.2 7.394 6.67 o81935245.8 ~0.82 @ between 2 and 6 customers exerusive leave in one 30 minutes period. solution: 1 Here, y ~ poissoni 2+2) and 6 Here: term exclusive means 2 are not involve here. i. e, PlzLY26) = P(4-3) + Ply=4) +P 14:5). 41 te x6.215 5! = 6-2-2 & 3. 1802 = 0.352376204 x 0.35

& in a at most 12 customers arrive 1 hour period. solution: For 30 minutes the mean pate of arriveis 4.5 u 1 minutes - - - - - a 60 minutes. - 4.560 4.560 so, X . [For Poisson (9) 1 hours period. I n - P.( X = 12) = P(x0) + P(x-1) +P 64-2) + P(x-3) + P(x-7) + P(x =5) + RCX6) + P(x-1) + P(X=8)+ P(x=0) - PCX= (0) + PC x = 1)) + P(x = 12 ot r o + e-9 31 te $? t ė -9 + 912 o - e-9 x 7096.4655 = 0.8757739

@ between 5 and 7 customers inclusive leave in one and half hour period Solution Here Y a poisson (2.2) 5 and t Here are in elusive means included i P(5.427). = P(y =5) + P(y =6) + P (Y=7). 5! 6! - .2 2 x-2-2 x 0.6 364 3265 x 0.6 36 4 3 265. = 0.07 0 5 18747