Question

Paul Hunt is considering two business ventures. The anticipated returns (in thousands of dollars) of each venture are described by the following probability distributions. Venture A Earnings -20 50 60 Probability 0.4 0.3 0.3 Venture B Earnings -15 20 50 Probability 0.2 0.5 0.3 (a) Compute the mean and variance for Venture A. mean 25000 dollars variance 660 * dollars Compute the mean and variance for Venture B. mean 22000 dollars variance 197.25 X dollars2 (b) Which investment would provide Paul with the highest expected return (the greater mean)? • Venture A Venture B (c) In which investment would the element of risk be less (that is, which probability distribution has the smaller variance)? O Venture A • Venture B

Answer 1

TOPIC:Expectation and variance of random variables.

= For ventune - A :- earnings Denote the dollaps) by x. [The question & Fon ventrine - A - Denote the earnings by x. is in dollars) ( where, x Earnings probability, P(x) The pmf of x is - Earnings () - 20,000 50,000 0.4 • 0.3 x is in dollars and in the question, it is given in thousands of dollars). 60,000 0.3 C So E(X). = mean. = I x. P(x). = - 20,000 x0.4 + 50,000x0.3 + 60,000 x 0.3. 25 000 domars

Again, E (+2), = I x².P (2). = (- 20,000) ² x 0.4 + (50,000" x 0.3 + C60,000) x 0.3. = 1990000000 dollars?, So, var (x) = E(x2) - E²(x). = 1990000000 - (25000) = [ 1365000000 dollars ² Ron venture -B :- Similarly, denote the earnings by the random variable, y (in dollars). So, the purf of y is - Earnings (Y) I probability, P(Y) -15,000 0.2 20,000 450,000 0.3 0.5 So, & (Y) = { J. P (7). = mean.

- 15000 x 0.2 + 20000 X 0.5 + 950 000x 0. 3 = 122000 dollars, Again, E (72) = 2 y? P(Y). = (-15000² * 0.2 + (2000) * xo. 5 + (50000² x 0. 3. 995000000 dollars So, var (y) = E(2) – E² (7) - 995000000 -(22000)? = 511 000 000 dollars since, E(X) Y E(Y) ; Venture A would provide C Paul with the highest expected return. since, var (x) > var (7) ; venture B souted will have less nisk. Tans