Question

Number of Food Number of Service Probabilit 0.12 0.29 0.42 0.17 laints 0.18 0.38 0.34 0.10 4.83 Refer to the informati on in th e previous exercise Find the mean and standard deviation of the to- tal number of complaints received in a week. Hav- ing reached this point, you are concerned that the numbers of food and service complaints may not be independent of each other. However, you have no information about the nature of their dependence What can you now say about the mean and standard deviation of the total number of complaints received in a week?

Answer 1

Solution:

Let random variable Z is the linear combination of two independent random variables X and Y.

then,

Z=aX+bY

Mean(μ_z)=aμ_x+μ_y

Variance(V_z)=a²σ²_x+b²σ²_y

X	P(x)	X*P(x)	X2*P(x)	Y	P(y)	Y*P(Y)	Y2*P(Y)
0	0.12	0	0	0	0.18	0.0	0
1	0.29	0.29	0.29	1	0.38	0.38	0.38
2	0.42	0.84	1.68	2	0.34	0.68	1.36
3	0.17	0.51	1.53	3	0.10	0.30	0.9
Total	1.0	1.64	3.50	Total	1.0	1.36	2.64