Question

3. (30 points) A shipping company offers the following plan: for any packaged to be shipped, there is a fixed $2.00 fee plus $2.50 per pound. Suppose that the company has a cost of $1.50 per pound. If the weight (rounded up) in pounds of the shipment is a random variable with the following p.m.f a) [8 points) Define Y profit. Describe the probability distribution of Y. b) [8 points] 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 +3.00

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

Y = X+3.00

Therefore,

P(Y=k+3.00) = k/15 for k =1,2,3,4,5

let l = k+3; k =l-3

Probability distribution of Y :

P(Y=l) = (l-3)/15 for l =4,5,6,7,8

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)
4	1/15
5	2/15
6	3/15
7	4/15
8	5/15

Expected Profit = E(Y)

$\small E(Y) = \sum l.P(Y=l) =4\times 1/15+5\times 2/15+6\times 3/15+7\times 4/15+8\times 5/15$

$\small =(1/15)(4+10+18+28+40)=100/15=6.67$

Expected profit E(Y) = 6.67

Expected profit = 6.67