Question

A shipping company offers the following plan: for any package to be shipped, they charge a fixed $2.00 fee in addition to a cost of $2.50 per pound and suppose that the company has a loss of $1.50 per pound. If the weight (rounded up) in pounds of the shipment is a random variable with the following probability mass function

P(X = k) = 15

k=1,2,3,4,5

a) Define Y = Profit. Describe the probability distribution of Y.

b) Find the expected profit of the company E(Y)

Answer 1

(a) X : Weight in pounds of the shipment.

Fixed Fee : $2.00

Per pound :$2.50

So total Fee for 'k' pounds = 2.00+2.5 k

Cost : $1.50 per pound

For k Pounds Cost = 1.5 k

Total fee for 'k' pounds -Cost for k pounds = 2.00 + 2.5k - 1.5k = k +2.00

Y = Profit ; and if X is the weight in pounds then

Y = X+2.00

Therefore,

P(Y=k+2.00) = k/15 for k =1,2,3,4,5

let l = k+2; k =l-2

Probability distribution of Y :

P(Y=l) = (l-2)/15 for l =3,4,5,6,7

Let us tabulate the pdf of X;

k:Pounds	P(X=k)
1	1/15
2	2/15
3	3/15
4	4/15
5	5/15

b.

l: Profit : $	P(Y=l)
3	1/15
4	2/15
5	3/15
6	4/15
7	5/15

Expected Profit = E(Y)

Expected profit E(Y) = 5.67

Expected profit = 5.67