From sample mean to expectation
For observations \(x_i\) with frequencies \(f_i\) in a sample of size \(n=\sum_i f_i\), the sample mean is
\[\bar x=\frac{\sum_i f_ix_i}{\sum_i f_i}=\frac{\sum_i f_ix_i}{n}=\sum_i x_i\frac{f_i}{n}\]
The quantity \(\frac{f_i}{n}\) is the relative frequency of \(x_i\). As \(n\to\infty\),
\[\lim_{n\to\infty}\frac{f_i}{n}=P(X=x_i)\]
The population mean, expectation or expected value of \(X\) is therefore its probability-weighted average.
\[\mu=E(X)=\sum_{\text{all}\hspace{0.5em}x}xP(X=x)\]
\(E(X)\) need not be one of the possible values of \(X\). It is a measure of central tendency, indicating the central value of the distribution.
Properties of expectation
If \(f\) is a real-valued function, then \(f(X)\) is also a discrete random variable and
\[E(f(X))=\sum_{\text{all}\hspace{0.5em}x}f(x)P(X=x)\]
\[E(a)=a,\qquad E(aX\pm b)=aE(X)\pm b\]
Need help? Join our JC Math tuition classes.
Learn more