4.1 Five of ten people earn $0, four earn $100, and one loses $100. What is the

Question

4.1 Five of ten people earn $0, four earn $100, and one loses $100. What is the expected payoff? What is the variance of the payoff?

4.2 There is a 50 percent chance of making $0, a 40 percent chance of making $100, and a 10 percent chance of losing $100. Calculate the expected value and variance of the payoff. How does your estimate compare to the previous problem?

4.3 There is a 1 percent chance that you will have healthcare bills of $100,000, a 19 percent chance that you will have healthcare bills of $10,000, a 60 percent chance that you will have healthcare bills of $500, and a 20 percent chance that you will have healthcare bills of $0. What is your expected healthcare spending?

Explanation / Answer

Ans)

4.1)

Let X denote the earnings

Then expected pay-off per person, E[X] = (5*$0 + 4*$100 + 1*(-100)) /10

E[X] = $300/10

E[X] = $30

Variance = pi * (X – mean)2

Variance of pay-off =0.5 (0-30)2 +0.4 (100 – 30)2 + 0.1(-100-30)2

Variance of pay-off = 4100

4.2)

Let Y denote the earnings and p denote the probability of the earnings.

Then expected pay-off of the individual, E[Y] = Y1*p1 + Y2*p2 +Y3*p3

E[Y] = $0*0.50 + $100*0.40 +(-$100)*0.10

E[Y] = $30

Variance of pay-off =0.50* (0-30)2 +0.40 (100 – 30)2 + 0.10(-100-30)2

Variance of pay-off = 4100

The expected pay-offs and variance are same for (a) and (b) as the earnings and the weights attached to these earnings are the same.

4.3)

Let Z denote the healthcare bills and p denote the percent chance of the healthcare bills.

Then expected healthcare spending, E[Z] = Z1*p1 + Z2*p2 +Z3*p3 + +Z4*p4

E[Z] = $100000*0.01 + $10000*0.19 +$500*0.60 + $0*0.20

E[Z] = $3200

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