A consultant is beginning work on three projects. The...

A consultant is beginning work on three projects. The expected profits from these projects are $$\$ 50,000$$, $$\$ 72,000$$, and $$\$ 40,000$$. The associated standard deviations are $$\$ 10,000, \$ 12,000$$, and $$\$ 9,000$$. Assuming independence of outcomes, find the mean and standard deviation of the consultant's total profit from these three projects.

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Key Concepts

Expected Value

The expected value is a fundamental concept in probability and statistics representing the average or mean outcome of a random variable. When combining multiple projects or random variables, the total expected value is the sum of the individual expected values, owing to the principle of linearity of expectation, which holds regardless of the independence of the variables.

Variance and Standard Deviation

Variance measures the spread or dispersion of a set of values from their mean, while the standard deviation is the square root of the variance. For independent random variables, the total variance is obtained by adding the individual variances. The standard deviation of the sum is then determined by taking the square root of this total variance, providing a measure of total risk or uncertainty.

Independence

Independence between random variables implies that the outcome of one does not affect the outcome of another. This property is critical when summing variances since it ensures that the covariance terms are zero, thereby validating the straightforward addition of individual variances to obtain the overall variance.

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