Questions: The probabilities are 0.15, 0.35, 0.2, and 0.3 that an investor will be able to sell a piece of property at a profit of 5,000, that he will be able to sell it at a profit of 3,000, that he will break even, or that he will sell it at a loss of 2,000. What is his expected profit?

Solution Steps

To find the expected profit, we use the formula for the mean of a discrete probability distribution:

\[ \text{Mean} = \sum (x_i \cdot p_i) \]

where \(x_i\) are the possible outcomes and \(p_i\) are the corresponding probabilities. Substituting the values:

\[ \text{Mean} = 5000 \times 0.15 + 3000 \times 0.35 + 0 \times 0.2 + (-2000) \times 0.3 \]

Calculating each term:

\[ = 750 + 1050 + 0 - 600 = 1200.0 \]

Thus, the expected profit is:

\[ \text{Expected Profit} = 1200.0 \]

The variance is calculated using the formula:

\[ \text{Variance} = \sigma^2 = \sum ((x_i - \text{Mean})^2 \cdot p_i) \]

Substituting the values:

\[ \text{Variance} = (5000 - 1200.0)^2 \times 0.15 + (3000 - 1200.0)^2 \times 0.35 + (0 - 1200.0)^2 \times 0.2 + (-2000 - 1200.0)^2 \times 0.3 \]

Calculating each term:

\[ = (3800)^2 \times 0.15 + (1800)^2 \times 0.35 + (1200)^2 \times 0.2 + (-3200)^2 \times 0.3 \]

\[ = 14440000 \times 0.15 + 3240000 \times 0.35 + 1440000 \times 0.2 + 10240000 \times 0.3 \]

\[ = 2166000 + 1134000 + 288000 + 3072000 = 6660000.0 \]

Thus, the variance is:

\[ \text{Variance} = 6660000.0 \]

The standard deviation is the square root of the variance:

\[ \text{Standard Deviation} = \sigma = \sqrt{6660000.0} \approx 2580.698 \]

Final Answer