Variance as Expectation of Square minus Square of Expectation/Continuous
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Theorem
Let $X$ be a continuous random variable.
Then the variance of $X$ can be expressed as:
- $\var X = \expect {X^2} - \paren {\expect X}^2$
That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$.
Proof
Let $\mu = \expect X$.
Let $X$ have probability density function $f_X$.
As $f_X$ is a probability density function:
- $\ds \int_{-\infty}^\infty \map {f_X} x \rd x = \Pr \paren {-\infty < X < \infty} = 1$
Then:
\(\ds \var X\) | \(=\) | \(\ds \expect {\paren {X - \mu}^2}\) | Definition of Variance of Continuous Random Variable | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{-\infty}^\infty \paren {X - \mu}^2 \map {f_X} x \rd x\) | Definition of Expectation of Continuous Random Variable | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{-\infty}^\infty \paren {x^2 - 2 \mu x + \mu^2} \map {f_X} x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_{-\infty}^\infty x^2 \map {f_X} x \rd x - 2 \mu \int_{-\infty}^\infty x f_X \paren x \rd x + \mu^2 \int_{-\infty}^\infty \map {f_X} x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X^2} - 2 \mu^2 + \mu^2\) | Definition of Expectation of Continuous Random Variable | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X^2} - \mu^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X^2} - \paren {\expect X}^2\) |
$\blacksquare$