Question

I have trouble with question c

Three salespeople decide to pool their commissions. Alice makes an average commission of $375 with a standard deviation of $60. Bob makes an average commission of $325 with a standard deviation of $55. Carl makes an average commission of $285 with a standard deviation $50. Assume that the amount of each commission follows a normal distribution that is independent of the other two.

(a) Calculate the probability that Alice makes more than $300 in a month.

(b) Calculate the probability that the average of the three commissions is greater than $300 in a month.

(c) If Alice and Bob’s commissions were positively correlated would that increase or decrease the variance of the average? Justify your response.

Answer 1

a)

probability that Alice makes more than $300 in a month =P(X>300)=1-P(X<300)

=1-P(Z<(300-375)/60)=1-P(Z<-1.25)=1-0.1056 =0.8944

b)

mean value of average =(375+325+285)/3=328.33

and std deviation =(1/3)*(60²+55²+50²)^1/2 =31.84

hence  probability that the average of the three commissions is greater than $300

=1-P(X<300)=1-P(Z<(300-328.33)/31.84)=1-P(Z<-0.89)=1-0.1867 =0.8133

c)

as commissions were positively correlated then variace of average will increases ; as covariance will positively increase in the variance