A factory manager is considering whether to replace a...

A factory manager is considering whether to replace a temperamental machine. A review of past records indicates the following probability distribution for the number of breakdowns of this machine in a week. $$ \begin{array}{lccccc} \hline \text { Number of breakdowns } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Probability } & 0.10 & 0.26 & 0.42 & 0.16 & 0.06 \\ \hline \end{array} $$ a. Find the mean and standard deviation of the number of weekly breakdowns. b. It is estimated that each breakdown costs the company $$\$ 1,500$$ in lost output. Find the mean and standard deviation of the weekly cost to the company from breakdowns of this machine.

A factory manager is considering whether to replace a temperamental machine. A review of past records indicates the following probability distribution for the number of breakdowns of this machine in a week.
$$
\begin{array}{lccccc}
\hline \text { Number of breakdowns } & 0 & 1 & 2 & 3 & 4 \\
\hline \text { Probability } & 0.10 & 0.26 & 0.42 & 0.16 & 0.06 \\
\hline
\end{array}
$$
a. Find the mean and standard deviation of the number of weekly breakdowns.
b. It is estimated that each breakdown costs the company $$\$ 1,500$$ in lost output. Find the mean and standard deviation of the weekly cost to the company from breakdowns of this machine.

Key Concepts

Linear Transformation of Random Variables

When a random variable is scaled or shifted by a constant, its mean and standard deviation are also transformed. Specifically, multiplying a random variable by a constant scales the mean by that constant and the standard deviation by the absolute value of the constant. This concept is important when converting the outcomes of a variable (such as the number of breakdowns) to another metric (like financial cost).

Probability Distribution

A probability distribution is a statistical function that shows all possible values of a random variable and the likelihood that each value will occur. It is fundamental to understanding data behavior and is utilized to compute various measures such as the mean and standard deviation, which summarize the characteristics of the random variable.

Expected Value

The expected value, or mean, of a random variable provides a measure of the central tendency of its distribution. It is computed by taking the sum of the products of each outcome and its corresponding probability, and it represents the long-run average if the random process were repeated many times.

Variance and Standard Deviation

Variance measures the spread of a random variable's outcomes around its mean, while the standard deviation is the square root of the variance. These measures indicate the degree of dispersion in the data and are essential for assessing the variability or risk associated with the random variable.

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