SHOW WORK A trucking company estimates that its costs for th

**SHOW WORK**

A trucking company estimates that its costs for the coming year depend on the average price of oil and other random factors as described by the following equation:

Costs = $600,000 + 5000 P_oil + U,

where P_oil is a random variable that equals the average price per barrel of oil and U is a random variable that captures other uncertain events that can affect costs for the year.

In words, the equation implies that the company has fixed costs of $600,000, that it expects to purchase 5,000 barrels of oil at whatever the average price of oil turns out to be during the year, and that costs are affected by random factors other than oil prices (U).

Analyses of historical data provide the following information:

P_oil and U are uncorrelated.

a) What is the expected value of Costs?

b) What is the standard deviation of Costs?

c) If Costs are normally distributed, what is the probability that costs are greater than the expected value of costs?

d) Again assume that Costs are normally distributed. The CFO would like to know the value of costs, call it X, such that the probability that Costs exceed X is 0.05. In other words, what value of X satisfies the following statement: Prob(Costs > X ) = 0.05.

Solution

Expected values of costs = E(600000+5000P+U)

= 600000+5000E(P)+0

= 600000+250000 = 850000

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b) Var (costs) = Var(600000+5000P+U) = 0+5000^2(10)²+25000²

= 2500,000,000+625,000,000

std deviation= 55901.70

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If costs are normal, then costs > average is half of area = 0.50

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P(Costs >X) = 0.05

z = 1.645 (from table)

Hence costs = Mean +1.645(std dev) = 850000+1.645(55901.70)

= 859708.30

Thus prob cost >859708.30 is 0.05