Gain Communications sells aircraft communication units to bo

Gain Communications sells aircraft communication units to both military and the civilian markets. Next year’s sales depend on market conditions that cannot be predicted exactly. Gain follows the modern practice of using probability estimates of sales. The military division estimates its sales as follows:

            Units sold        1000                3000                5000                10,000

            Probability       0.1                   0.3                   .4                     .2

These are personal probabilities that express the informed opinion of Gain’s executives. The corresponding sales estimates for the civilian division are

                        Units sold        300                  500                  750

                        Probability       0.4                   0.5                   0.1

Take X to be the number of military units sold and Y the number of civilian units sold.

a)From the probability distribution compute the estimated average sales for military division and civilian division.

b)Gain makes a profit of $2000 on each military unit sold and $3500 on each Civilian unit. Calculate next year’s total profit for gain Communication.

c)Find the variance and standard deviation of military sales(X) and civilian sales(Y).

d)Because the military budget and the civilian economy are not closely linked, Gain is willing to assume its military and civilian sales vary independently. What is the standard deviation of Gain’s total sales(X+Y)?

e.)Find the standard deviation of estimated profit (Z = 2000X+3500Y

Solution

a)

Estimated average sales for military division = 0.1*1000+0.3*3000+0.4*5000+0.2*10000 = 4000

Estimated average sales for civilian division = 0.4*300 + 0.5*500+0.1*750 = 445

b)

Profit from military division sales = $ 2000*4000 = $8,000,000

Profit from civilian division sales = $3500*445 = $1,557,500

Total Profit = $8,000,000+$1,557,500 = $9,557,500

c)

Variance for military division = 0.1*(1000-4000)²+0.3*(3000-4000)²+0.4*(5000-4000)²+0.2*(10000-4000)² = 8,800,000

Variance for civilian division = 0.4*(300-445)² + 0.5*(500-445)²+0.1*(750-445)² = 19,225

Standard deviation of military division = sqrt(8,800,000) = 2966.5

Standard deviation for sales in civilian division = sqrt(19,225) = 138.6

d)

standard deviation of (X+Y) = sqrt(Var (X)+Var(Y)) = sqrt(8,800,000+19225) = sqrt(8,819,225) = 2969.7

e)

standard deviation of Z = sqrt(2000²var(X)+3500²Var(Y))= sqrt(2000²*8800000+3500²*19225) = 5,952,773