The total cost for a production process is equal to 1,000...

The total cost for a production process is equal to $$\$ 1,000$$ plus two times the number of units produced. The mean and variance for the number of units produced are 500 and 900 , respectively. Find the mean and variance of the total cost.

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Key Concepts

Linear Transformation of Random Variables

A linear transformation of a random variable involves applying a function of the form Y = a + bX, where a and b are constants and X is the original random variable. This transformation is useful because it allows us to calculate the new mean and variance directly from those of X using simple linear relationships.

Expected Value

The expected value or mean of a random variable is a measure of its central tendency, representing the average outcome if an experiment were repeated many times. Under a linear transformation, the new expected value is given by a + b times the original expected value, preserving the linearity of the transformation.

Variance

Variance quantifies the spread or dispersion of a random variable around its mean, indicating the variability of the data. In a linear transformation, while adding a constant does not affect the variance, scaling the variable by a constant multiplies the variance by the square of that constant. This property allows for straightforward calculation of the transformed variance.

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