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QUAN 2010 Notes Introduction to Business Statistics

Lauren M. Nelsen and Kimberly Swetz

Section 5.2 Descriptive Statistics for a Discrete Probability Distribution

There are a lot of things we might want to ask when we’re working with a discrete probability distribution. For example, what average value would we expect rolling a die or playing a gambling game? How spread out are the values? These are all questions we want to be able to answer.
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Definition 5.2.1.

  • The mean, or expected value of a discrete probability distribution is the weighted average of all outcomes of the random variable.
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    \begin{equation*} E(X)=\mu=\sum_{i=1}^n x_iP(x_i) \end{equation*}
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  • The expected monetary value is the mean when the discrete random variable is in dollars ($)
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  • The variance of a discrete probability distribution measures the spread of the individual values around the mean.
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    \begin{equation*} \sigma^2=\sum_{i=1}^n (x_i-\mu)^2P(x_i) \end{equation*}
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  • The standard deviation of a discrete probability distribution measures the dispersion of the outcome of the random variable in relation to the mean.
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    \begin{equation*} \sigma=\sqrt{\sigma^2} \end{equation*}
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Example 5.2.2.

(Donnely 5.7) The table below shows the discrete probability distribution for the number of bedrooms per house in a certain community:
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\(\#\) of Bedrooms Probability
3 0.23
4 0.57
5 0.14
6 ???

(a)

Determine the missing probability for a 6-bedroom house.
Answer.
\begin{equation*} 1-(0.23+0.57+0.14)=0.06 \end{equation*}
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(b)

Determine the mean number of bedrooms per house.
Answer.
\begin{equation*} \mu=\text{expected number of bedrooms}=3\cdot (0.23)+4\cdot (0.57)+5\cdot (0.14)+6\cdot (0.06)=4.03 \end{equation*}
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(c)

Determine the standard deviation for the number of bedrooms per house. (Hint: Use a table like the one below in Excel to compute the variance first. Do all the calculations in Excel.)
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Here is a link to an Excel file with a blank table like the one below: external/sheets/BedroomsCalculation.xlsx
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Table 5.2.3. Calculation of Standard Deviation
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\(x\) \(P(x)\) \(\mu\) \(x-\mu\) \((x-\mu)^2\) \((x-\mu)^2P(x)\)
3
4
5
6
Answer.
Standard deviation: \(\approx 0.7804\)
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Here is an Excel file where you can see the work for this problem: external/sheets/BedroomsCalculationSolution.xlsx
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Example 5.2.4.

(Donnelly 5.45)
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Tees R Us manufactures and sells T-shirts for sporting events, is providing shirts for an upcoming tournament. Each shirt will cost \(\$9\) to produce and will be sold for \(\$18\text{.}\) Any unsold shirts at the end of the tournament can be sold for \(\$5\) a piece in the near future. Tees R Us assumes the demand for the shirts will be \(500\text{,}\) \(1000\text{,}\) \(1500\text{,}\) or \(2000\text{.}\) They also estimate that the probabilities of each of these sales levels occurring will be \(15\%\text{,}\) \(20\%\text{,}\) \(25\%\text{,}\) and \(40\%\text{,}\) respectively. Determine the expected monetary value of the project if Tees R Us chooses to print \(1500\) shirts for the tournament.
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Here is an Excel file you can use for this problem: external/sheets/TeesRUs.xlsx
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Answer.
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