Editing Variance (section)

==Examples==

===Exponential distribution===
The [[exponential distribution]] with parameter {{mvar|&lambda;}} > 0 is a continuous distribution whose [[probability density function]] is given by
<math display="block">f(x) = \lambda e^{-\lambda x}</math>
on the interval {{closed-open|0, ∞}}.  Its mean can be shown to be
<math display="block">\operatorname{E}[X] = \int_0^\infty x \lambda e^{-\lambda x} \, dx = \frac{1}{\lambda}.</math>

Using [[integration by parts]] and making use of the expected value already calculated, we have:
<math display="block">\begin{align}
  \operatorname{E}\left[X^2\right] &= \int_0^\infty x^2 \lambda e^{-\lambda x} \, dx \\
    &= {\left[ -x^2 e^{-\lambda x} \right]}_0^\infty + \int_0^\infty 2 x e^{-\lambda x} \,dx \\
    &= 0 + \frac{2}{\lambda}\operatorname{E}[X] \\
    &= \frac{2}{\lambda^2}.
\end{align}</math>

Thus, the variance of {{mvar|X}} is given by
<math display="block">\operatorname{Var}(X) = \operatorname{E}\left[X^2\right] - \operatorname{E}[X]^2 = \frac{2}{\lambda^2} - \left(\frac{1}{\lambda}\right)^2 = \frac{1}{\lambda^2}.</math>

===Fair die<!--Singular: die; plural: dice. Don't change-->===
A fair [[dice|six-sided die]] can be modeled as a discrete random variable, {{mvar|X}}, with outcomes 1 through 6, each with equal probability 1/6.  The expected value of {{mvar|X}} is <math>(1 + 2 + 3 + 4 + 5 + 6)/6 = 7/2.</math> Therefore, the variance of {{mvar|X}} is
<math display="block">\begin{align}
  \operatorname{Var}(X) &= \sum_{i=1}^6 \frac{1}{6}\left(i - \frac{7}{2}\right)^2 \\[5pt]
    &= \frac{1}{6}\left((-5/2)^2 + (-3/2)^2 + (-1/2)^2 + (1/2)^2 + (3/2)^2 + (5/2)^2\right) \\[5pt]
    &= \frac{35}{12} \approx 2.92.
\end{align}</math>

The general formula for the variance of the outcome, {{mvar|X}}, of an {{nowrap|{{mvar|n}}-sided}} die is
<math display="block">\begin{align}
  \operatorname{Var}(X) &= \operatorname{E}\left(X^2\right) - (\operatorname{E}(X))^2 \\[5pt]
    &= \frac{1}{n}\sum_{i=1}^n i^2 - \left(\frac{1}{n}\sum_{i=1}^n i\right)^2 \\[5pt]
    &= \frac{(n + 1)(2n + 1)}{6} - \left(\frac{n + 1}{2}\right)^2 \\[4pt]
    &= \frac{n^2 - 1}{12}.
\end{align}</math>

=== Commonly used probability distributions ===
The following table lists the variance for some commonly used probability distributions.
{| class="wikitable"
|-
! Name of the probability distribution
! Probability distribution function
! Mean
! Variance
|-
| [[Binomial distribution]]
| <math>\Pr\,(X=k) = \binom{n}{k}p^k(1 - p)^{n-k}</math>
| <math>np</math>
! <math>np(1 - p)</math>
|-
| [[Geometric distribution]]
| <math>\Pr\,(X=k) = (1 - p)^{k-1}p</math>
| <math>\frac{1}{p}</math>
! <math>\frac{(1 - p)}{p^2}</math>
|-
| [[Normal distribution]]
| <math>f\left(x \mid \mu, \sigma^2\right) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{1}{2} {\left(\frac{x - \mu}{\sigma}\right)}^2}</math>
| <math>\mu</math>
! <math>\sigma^2</math>
|-
| [[Uniform distribution (continuous)]]
| <math>f(x \mid a, b) = \begin{cases}
  \frac{1}{b - a} & \text{for } a \le x \le b, \\[3pt]
                0 & \text{for } x < a \text{ or } x > b
  \end{cases}</math>
| <math>\frac{a + b}{2}</math>
! <math>\frac{(b - a)^2}{12}</math>
|-
| [[Exponential distribution]]
| <math>f(x \mid \lambda) = \lambda e^{-\lambda x}</math>
| <math>\frac{1}{\lambda}</math>
! <math>\frac{1}{\lambda^2}</math>
|-
| [[Poisson distribution]]
| <math>f(k \mid \lambda) = \frac{e^{-\lambda}\lambda^{k}}{k!}</math>
| <math>\lambda </math>
! <math>\lambda </math>
|}