Question

uestion 5 (I mark) Afer analyzing several months of sales data, the owner of an appliance store produced the followin joint probability distribution of the number of refrigerators and stoves sold hourly Stoves Refrigerators 0 0 0.14 0.12 0.09 0.17 0.13 0.05 0.27 0) 0.29 a) Fill in the missing values above. b) What are the laws for a discrete probability density function? c) If a customer purchases 2 stoves, what is the probability that they also purchase 2 refrigerators? d) What is the average number of refrigerators purchased? e) What is the variance in the number of refrigerators purchased? f) Are the sales of stoves and refrigerators independent? g) What is the marginal probability distribution for sales in refrigerators if the customer did not purchase stove? What is the expected value and variance for sales in refrigerators, if the customer did not purchase a stove? The part of the original chart we use to get the solution: Stoves Refrigerators 0 0 0.14 0.12 Stoves Refrigerators R OSCR1SR-2t.o If a "sale" can be either a stove or a fridge, what is the average number of sales? Is the variance in "sales" smaller or larger than the variance in stoves? (Carefully state any assumptions.) h)

Answer 1

a) Given table A B C D E Stoves/ fs(i) 0 1 2 1 Refrigerators 0.14 0.17 iv) (vii) 0.12 0.13 v) 0.29 |(ii) (ii) 0.27 |(viii)=1 2 0 0.09 0.05 |vi) 3 1 4 2 5 fR(i) Table 1.1 shows the filled table: Table 1.1 B C D E Stoves/ 1 2 fs(i) 1 Refrigerators 0.08 0.09 0.05 0.22 0.12 0.13 0.34 0.14 0.17 0.18 0.49 2 0 0.39 0.27 3 1 4 2 0.04 0.29 5 fR(i) Table 1.2 shows the used formula to calculate the values of Table 1.1 Table 1.2 B C D E fsi) Stoves/ Refrigerators 0 2 0.12 0.13 -(E4- (B4+ D4 ) -(D5-(D2+ D3) ) | 0.27 1 -(E2- ( C2+ D2 ) ) 0.14 0.09 0.05 (B2+B3+B4 ) +(C2+C3+C4 ) 0.29 |=(E5-(E3+E4)) ЕВз-С3+D3) 2 0 0.17 3 l 4 2 1 5 fR(i)

b) Laws for a discreate probability density function: ViJ.P(X,,Y,)e P(X =i)=EP(X=i,Y = j) j-0 c) Calculate the probability that customer also purchase 2 refrigerators: P[(R,S)=(2,2)]0.04 P(S =2) 0.148 0.27 d) Calculate the average number of refrigerators purchased: E(R)=EPR, = (0x0.22)+(1×0.49)+(2×0.29) = 0+0.49+0.58 1.07 %3D e) Calculate the variance in the number of refrigerators purchased: E(R*)-EPR -(0* x0.22)-(1* x0.49)+(2° x0.29) = 0+0.49+1.16 1.65 f) Are the sales of stoves and refrigerators independent? P(R=0)=0.22, P(S =0)= 0.34, P(R=0& S = 0)=0.08 :P(R=0& S =0) P(R=0)xP(S%=D0) Thus, it is not independent