# Expected Value, Variance, Binomial Distribution

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1) A $25,000 investment in a tract of land may be worth $10,000, $25,000, or $45,000 after one year, the probabilities of these values being 0.25, 0.45, and 0.3, respectively.

a) What is the expected value of the investment in one year?

b) If your expected return on the investment is the expected present value of the investment minus the amount you invested and the discount rate is 0.04, what is the value of your expected return? (Recall: If you have an investment that is worth $25,000 one year from now, the present value of that investment is $(25,000/1.04) = $24038.46.)

2) A discrete random variable U has the following probability distribution:

Value of U 0 1 4 9 16

PU(u) 0.2 0.25 0.35 0.1 0.1

a) Find E(U), E(U2), and E(U½).

b) Using the results of (a), find E(U2 - U), [E(U2 - 4U)]½ , and E(U - U½ + 4) .

3) In an article in the New York Times magazine entitled "The Great Oil Rush of the Eighties," an oil company official was quoted as saying, "Our odds of success on a wildcat well in the Overthrust are about a hundred to one." Of course, even if oil is found, the amount can vary greatly. For simplicity, assume that it costs $1 million to drill a well and that, if oil is found, the net return (gross return minus costs) will be $199 million. Let R denote the net return when a wildcat well is drilled in this area. If the probability of hitting a dry well (no oil) is 0.99, find

a) the expected value of the net return,

b) the variance of the net return, and

c) the standard deviation of the net return.

4) Seth can rent an ice cream push cart for $75 per day. Dry ice for the freezer costs an additional $35 per day. He makes $0.40 on each ice cream sale, but the number sold is a random variable having a mean of 300 per day and a standard deviation of 20 per day. Find

a) his expected daily profit,

b) the variance of his daily profit, and

c) the standard deviation of his daily profit.

5) Using Table 2 in the text, find the following, when X is a binomial random variable with the given parameters:

a) P(X = 0), when n = 10 and = 0.2.

b) P(X 1), when n = 6 and = 0.4.

c) P(X = 5), when n = 15 and = 0.75.

d) P(X 14), when n = 19 and = 0.65.

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The solution provides answers to 5 questions on expected value, binomial distribution, variance, standard deviation, probability.

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The answers are in the excel sheets.

1) A $25,000 investment in a tract of land may be worth $10,000, $25,000, or $45,000 after one year, the probabilities of these values being 0.25, 0.45, and 0.3, respectively.

a) What is the expected value of the investment in one year?

Prob Value Prob* Value

0.25 10000 2500

0.45 25000 11250

0.3 45000 13500

1 Σ= 27250

Expected value of investment= 27250

b) If your expected return on the investment is the expected present value of the investment minus the amount you invested and the discount rate is 0.04, what is the value of your expected return? (Recall: If you have an investment that is worth $25,000 one year from now, the present value of that investment is $(25,000/1.04) = $24038.46.)

Discount rate= 0.04

Expected value of investment=$(27,250/1.04) = 26202

expected return on the investment=26202-25000 1202

2) A discrete random variable U has the following probability distribution:

Value of U 0 1 4 9 16

PU(u) 0.2 0.25 0.35 0.1 0.1

PU(u) Value of ...

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