A company specializes in installing and servicing...

A company specializes in installing and servicing central-heating furnaces. In the prewinter period, service calls may result in an order for a new furnace. The following table shows estimated probabilities for the numbers of new furnace orders generated in this way in the last two weeks of September. $$ \begin{array}{lcccccc} \hline \text { Number of orders } & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Probability } & 0.10 & 0.14 & 0.26 & 0.28 & 0.15 & 0.07 \\ \hline \end{array} $$ a. Graph the probability distribution function. b. Calculate and graph the cumulative probability distribution. c. Find the probability that at least 3 orders will be generated in this period. d. Find the mean of the number of orders for new furnaces in this 2 -week period. e. Find the standard deviation of the number of orders for new furnaces in this 2 -week period.

A company specializes in installing and servicing central-heating furnaces. In the prewinter period, service calls may result in an order for a new furnace. The following table shows estimated probabilities for the numbers of new furnace orders generated in this way in the last two weeks of September.
$$
\begin{array}{lcccccc}
\hline \text { Number of orders } & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Probability } & 0.10 & 0.14 & 0.26 & 0.28 & 0.15 & 0.07 \\
\hline
\end{array}
$$
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the probability that at least 3 orders will be generated in this period.
d. Find the mean of the number of orders for new furnaces in this 2 -week period.
e. Find the standard deviation of the number of orders for new furnaces in this 2 -week period.

Key Concepts

Probability Mass Function

A probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. It is typically represented in a table or graph form where each possible outcome is paired with its probability, allowing for a clear visualization of the distribution of outcomes.

Cumulative Distribution Function

The cumulative distribution function (CDF) for a discrete random variable shows the probability that the variable takes on a value less than or equal to a specific value. It is calculated by summing the probabilities from the PMF up to that value and is often graphically represented to illustrate how probability accumulates across the range of possible outcomes.

Computing Cumulative Probability for 'At Least' Events

When asked for the probability that the random variable exceeds a certain value (for instance, at least a certain number), the relevant value can be computed by either summing the probabilities for all outcomes at or above the threshold using the PMF or by using the CDF to subtract the cumulative probability below the threshold from 1. This approach is essential for understanding tail probabilities in the context of discrete events.

Expected Value

The expected value, or mean, of a discrete random variable is a measure of its central tendency and is computed as the sum of the products of each possible value of the variable with its corresponding probability. It gives a weighted average outcome and is foundational in assessing the typical value expected in repeated experiments.

Standard Deviation

The standard deviation is a measure of the dispersion or spread of a discrete random variable around its mean. It is calculated by taking the square root of the variance, where the variance is the weighted average of the squared differences between each value and the mean. This measure helps to understand the variability in the outcomes of the random variable.

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