The profit for a production process is equal to 1,000 minus...

The profit for a production process is equal to $$\$ 1,000$$ minus two times the number of units produced. The mean and variance for the number of units produced are 50 and 90 , respectively. Find the mean and variance of the profit.

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Key Concepts

Variance

Variance quantifies how spread out the outcomes of a random variable are around its mean. It indicates the degree of variability in the data. Under a linear transformation, the variance is scaled by the square of the coefficient multiplying the random variable, while any added constant does not affect the dispersion. This allows for straightforward computation of the new variance from the original.

Linear Transformation of Random Variables

When a random variable is transformed by a function of the form aX + b, the properties of the expected value and variance can be used to easily compute the new mean and variance. In particular, the expected value is given by aE(X) + b, and the variance is given by a²Var(X), with the constant term b having no effect on the variance.

Expected Value

The expected value measures the average outcome of a random variable over many trials. It is the long-run mean of the distribution and is calculated by summing the products of each possible outcome with its probability. For linear transformations, this property simplifies to multiplying the original mean by the coefficient and then adding the constant.

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