Definition
The mean of a discrete random variable, \(X\), is defined to be
\[ E[X] = \mu = \sum_{\text{all } x} {x \times p(x)} \]This corresponds closely to the definition of the mean of a discrete data set. For example, the following frequency table summarises the distribution of 600 discrete values.
Household size x |
Frequency ƒx |
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total | 600 |
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The mean of the data is
\[\begin{align} \overline{x} = \frac {\sum x} n & = \frac {\overbrace{1 + 1 + ... + 1}^{140} \; + \; \overbrace{2 + 2 + ... + 2}^{180} \; + \; \overbrace{3 + 3 + ... + 3}^{60} \; + \; ...} {600} \\ & = \frac {140 \times 1 \; + \; 180 \times 2 \; + \; 60 \times 3 \; + \; ...} {600} \\ & = \frac {140} {600} \times 1 \; + \; \frac {180} {600} \times 2 \; + \; \frac {60} {600} \times 3 \; + \; ... \\ & = \sum_{x=1}^7 {x \times \text{Propn}(x)} \\ & = 2.933 \end{align} \]If \(X\) is defined as a randomly chosen one of these 600 values, the probabilities of getting {1, ..., 7} would be their proportions in the data set and would have the same mean,
\[ \mu \;=\; \sum_{x=1}^7 {x \times p(x)} \;=\; 2.933 \]