Roy and Jennie are looking for a company to help plan their wedding.

Question

Company A charges an amount based on the cost function, C=Vn + F, where C is total cost, V is variable cost, F is fixed cost, and n is the number of people who attend the wedding. Suppose n is a random variable with mean=41, Standard deviation=9. V=40 and F=2950. (Round answers to nearest dollar)

a) What is the mean for the total cost?

b) What is the standard deviation for the total cost?

Answer 1

This problem is best approached by using the Linearity of Expectation and Linearity of Variance.

The Linearity of Expectation states that E(aX+b)=a*E(X)+b. So, in this problem, we want the expectation (mean) of E(C)=E(Vn+F)=V*E(n)+F. This is equal to (40*41), remember the mean or expectation of n is 41 and V=40, added to F=2950. Thus, the E(C)=4590.

The Linearity of Variance states that Var(aX+b)=a²*Var(X). So, in this problem, we want the standard deviation of C, the total cost. If we recall that the variance is simply the standard deviation squared, then we we know that the variance of n is 9²=81. With all of this in mind, we start with Var(C)=Var(Vn+F)=V²*Var(n). Thus, for the variance of the total cost we get 40²*81=129600. Now if we take the square root of this number, we will have our standard deviation of the total cost. So, SD(C)=sqrt(129600)=360.

Mean of total cost = 4590

Standard Deviation of total cost = 360

Linearity of Variance Proof (if needed):

If Y=aX+b, E(Y)=aE(X)+b, then:

Var(Y)=E[(Y-E(Y)²]

=E[aX+b-aE(X)+b)²]

=E[a²(X-µ_x)²]

=a²E[(X-µ_x)²]

Var(Y)=a²Var(X)