A gas station sells three of gassoline: regular, extra, and supper. These are pr

Question

A gas station sells three of gassoline: regular, extra, and supper. These are priced at $2.30, $ 2.85, $ 3.40 per gallon, respectively. Let X1, X2, X3 denote the amounts of these grades purchased(gallons) on a particular day. Suppose that Xi are independent with µ1 = 1000,µ2 = 500,µ3 = 300,1 = 100,2 = 80,3 = 50.

(a) What is the expression of the revenue from sales? [Hint: Y = 2.3X1 + 2.85X2 + 3.40X3

(b) Expected value of the revenue( E(Y )) ?

(c) Variance of the revenue (V (Y )) ?

(d) Standard deviation of the revenue ( Y ) ?

Explanation / Answer

(a) Here Y is the Revenue from sales.

Y= 2.3X1 + 2.85X2 + 3.40 X3

where X1 ~ N(1000, 100) , X2 ~ N(500,80) and X3 ~ N(300, 50)

(b) E(Y) = E( 2.3X1 + 2.85X2 + 3.40 X3 ) = 2.3 * 1000 + 2.85 * 500 + 3.40 * 300 = $ 4745

(c) Var(Y) = 2.32 Var(X1 ) + 2.852 Var(X2 ) + 3.402 Var(X3)

= (2.3 * 100)2 + (2.85 * 80)2 + (3.40 * 50)2

= 133784

(d) Standard deviation of Y = Y = sqrt(133784) = 365.765

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