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.
