Answer:
Expected number of sweaters per month can be given as follows:
E(X) = Σ x P(X = x)
Now,
E(X) = [2 * 0.1 + 3* 0.1+ 4* 0.2 + 5* 0.3 + 6* 0.2 + 7 * 0.1]
E(X) = 4.7.
Var(X) = E(X^2) – [E(X)]^2
We have E(X) = 4.7. Thus, [E(X)]2= 4.7*4.7 = 22.09.
Now E(X^2) = [2*2 * 0.1 + 3*3* 0.1+ 4*4* 0.2 + 5*5* 0.3 + 6*6* 0.2 + 7*7*0.1]
E(X^2) = 24.1
Thus by formula, Var(X) = E(X2) – [E(X)]2
Var(X) = 24.1-22.09
Var(X) = 2.01
Given that exporter pays the $12 for each sweater. The woman pays $2 per sweater. The cost of shipment is $3 irrespective of the number of sweaters. Now, let m is the number of sweaters she made. Thus, the total cost she would have to pay would be
Total cost by woman = 2m+3
The total cost paid by the exporter would be = 12m.
Now the profit of woman would be given by,
= The total cost exporter pay – cost paid by the woman
= 12m – (2m +3)
= 12m – 2m -3
= 10m – 3.
Now expected profit made by the woman is given in the following table below:
E(Profit) = Σ profit* P(X = x)
In a similar way, as we have done in part (a).
E(Profit) = [17 * 0.1 + 27* 0.1+ 37* 0.2 + 47* 0.3 + 57* 0.2 + 67 * 0.1]
E(Profit) = 44.
Now, we calculate the variance:
Var(profit) = E(profit^2) – [E(profit)]^2
Var(profit) =
E(profit^2) = [17*17 * 0.1 + 27*27* 0.1+ 37*37* 0.2 + 47*47* 0.3 + 57*57* 0.2 + 67*67*0.1]
E(profit^2) = 2137.
[E(profit)]^2 = 44*44 = 1936.
Thus, the variance can be given as =
Var(profit)= 2137 – 1936
Var(profit) = 201.
since dividing numbers with exponents would be subtracting the bottom exponent from the top exponent, provided that the base number (in this case 3) is the same for both. For this method, you would get 
, which is equal to 1/9 or .1 repeating, the same answer that you'd get with the first method.