Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ ex-Gaussian distribution ▷ Variance

Theorem: Let $X$ be a random variable following an ex-Gaussian distribution:

\[\label{eq:exg} X \sim \text{ex-Gaussian}(\mu, \sigma, \lambda) \; .\]

Then, the variance of $X$ is

\[\label{eq:exg-var} \mathrm{Var}(X) = \sigma^2 + \frac{1}{\lambda^2} \; .\]

Proof: To compute the variance of $X$, we partition the variance into expected values:

\[\label{eq:var-mean} \mathrm{Var}(X) = \mathrm{E}(X^2)-\mathrm{E}(X)^2 \; .\]

We then use the moment-generating function of the ex-Gaussian distribution to calculate

\[\label{eq:exg-moment} \mathrm{E}(X^2) = M_X''(0)\]

based on the relationship between raw moment and moment-generating function.

First, we differentiate

\[\label{eq:exg-mgf} M_X(t) = \left( \frac{\lambda}{\lambda-t} \right) \exp \left[ \mu t + \frac{1}{2}\sigma^2t^2 \right]\]

with respect to $t$. Using the product rule and chain rule gives:

\[\label{eq:exg-var-s1} \begin{split} M'_X(t) &= \frac{\lambda}{(\lambda-t)^2}\exp \left[ \mu t + \frac{1}{2}\sigma^2t^2\right] + \left(\frac{\lambda}{\lambda-t}\right)\exp\left[\mu t + \frac{1}{2}\sigma^2t^2\right] (\mu + \sigma^2t)\\ &= \left(\frac{\lambda}{\lambda-t}\right) \cdot \exp\left[\mu t + \frac{1}{2}\sigma^2t^2\right] \cdot \left[ \frac{1}{\lambda-t} +\mu +\sigma^2t \right] \\ &= M_X(t)\cdot \left[ \frac{1}{\lambda-t} + \mu + \sigma^2t\right] \; . \end{split}\]

We now use the product rule to obtain the second derivative:

\[\label{eq:exg-var-s2} \begin{split} M_X''(t) &= M_X'(t)\cdot \left[ \frac{1}{\lambda-t} + \mu + \sigma^2t\right] + M_X(t)\cdot \left[ \frac{1}{(\lambda-t)^2}+\sigma^2\right] \\ &\overset{\eqref{eq:exg-var-s1}}{=} M_X(t)\cdot \left[ \frac{1}{\lambda-t} + \mu + \sigma^2t\right]^2 + M_X(t)\cdot \left[ \frac{1}{(\lambda-t)^2}+\sigma^2\right]\\ &= M_X(t) \cdot \left[ \left( \frac{1}{\lambda-t} + \mu + \sigma^2t\right)^2 + \frac{1}{(\lambda-t)^2} + \sigma^2\right] \\ &= \left(\frac{\lambda}{\lambda-t}\right)\cdot \exp\left[\mu t + \frac{1}{2}\sigma^2t^2\right]\cdot \left[ \left( \frac{1}{\lambda-t} + \mu + \sigma^2t\right)^2 + \frac{1}{(\lambda-t)^2} + \sigma^2\right] \end{split}\]

Applying \eqref{eq:exg-moment} yields

\[\label{eq:exg-var-s3} \begin{split} \mathrm{E}(X^2) &= M_X''(0) \\ &= \left(\frac{\lambda}{\lambda-0}\right) \cdot \exp\left[\mu\cdot 0 + \frac{1}{2}\sigma^2\cdot 0^2\right] \cdot \left[\left(\frac{1}{\lambda-0}+\mu + \sigma^2\cdot 0\right)^2 + \frac{1}{(\lambda-0)^2}+\sigma^2\right]\\ &= 1\cdot 1 \cdot \left[\left(\frac{1}{\lambda} + \mu\right)^2 + \frac{1}{\lambda^2}+\sigma^2 \right]\\ &= \frac{1}{\lambda^2} + \frac{2\mu}{\lambda} + \mu^2 + \frac{1}{\lambda^2} + \sigma^2\\ &= \frac{2}{\lambda^2} + \frac{2\mu}{\lambda} + \mu^2 + \sigma^2 \; . \end{split}\]

Since the mean of an ex-Gaussian distribution is given by

\[\label{exg-mean} \mathrm{E}(X) = \mu + \frac{1}{\lambda} \; ,\]

we can apply \eqref{eq:var-mean} to show

\[\label{eq:exg-var-s4} \begin{split} \mathrm{Var}(X) &= \mathrm{E}(X^2) - \mathrm{E}(X)^2 \\ &= \left[\frac{2}{\lambda^2} + \frac{2\mu}{\lambda} + \mu^2+\sigma^2\right] - \left(\mu+\frac{1}{\lambda}\right)^2\\ &= \frac{2}{\lambda^2} + \frac{2\mu}{\lambda} + \mu^2+\sigma^2 - \mu^2 - \frac{2\mu}{\lambda} - \frac{1}{\lambda^2}\\ &= \sigma^2+\frac{1}{\lambda^2} \; . \end{split}\]

This completes the proof of \eqref{eq:exg-var}.

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Metadata: ID: P406 | shortcut: exg-var | author: tomfaulkenberry | date: 2023-04-19, 12:00.