Editing Mixture distribution (section)

=== Moments ===
Let {{math|''X''<sub>1</sub>}}, ..., {{math|''X''<sub>''n''</sub>}} denote random variables from the {{mvar|n}} component distributions, and let {{mvar|X}} denote a random variable from the mixture distribution. Then, for any function {{math|''H''(·)}} for which <math>\operatorname{E}[H(X_i)]</math> exists, and assuming that the component densities {{math|''p<sub>i</sub>''(''x'')}} exist,

<math display="block">\begin{align}
\operatorname{E}[H(X)] & = \int_{-\infty}^\infty H(x) \sum_{i = 1}^n w_i p_i(x) \, dx \\
& = \sum_{i = 1}^n w_i \int_{-\infty}^\infty p_i(x) H(x) \, dx = \sum_{i = 1}^n w_i \operatorname{E}[H(X_i)].
\end{align}</math>

The {{mvar|j}}th moment about zero (i.e. choosing {{math|1=''H''(''x'') = ''x''{{i sup|''j''}}}}) is simply a weighted average of the {{mvar|j}}-th moments of the components.  Moments about the mean {{math|1=''H''(''x'') = (''x − μ''){{i sup|''j''}}}} involve a binomial expansion:<ref>{{harvtxt|Frühwirth-Schnatter|2006|at=Ch.1.2.4}}</ref>

<math display="block">\begin{align}
\operatorname{E}\left[{\left(X - \mu\right)}^j\right]
& = \sum_{i=1}^n w_i \operatorname{E}\left[{\left(X_i - \mu_i + \mu_i - \mu\right)}^j\right] \\
& = \sum_{i=1}^n w_i \sum_{k=0}^j \binom{j}{k} {\left(\mu_i - \mu\right)}^{j-k} \operatorname{E}\left[{\left(X_i - \mu_i\right)}^k\right],
\end{align}</math>

where {{math|''μ<sub>i</sub>''}} denotes the mean of the {{mvar|i}}-th component.

In the case of a mixture of one-dimensional distributions with weights {{math|''w<sub>i</sub>''}}, means {{math|''μ<sub>i</sub>''}} and variances {{math|''σ''<sub>''i''</sub><sup>2</sup>}}, the total mean and variance will be:
<math display="block"> \operatorname{E}[X] = \mu = \sum_{i = 1}^n w_i \mu_i ,</math><math display="block"> \begin{align}
\operatorname{E}\left[(X - \mu)^2\right] & = \sigma^2 \\
& = \operatorname{E}[X^2] - \mu^2 & (\text{standard variance reformulation})\\
& = \left(\sum_{i=1}^n w_i \operatorname{E}\left[X_i^2\right]\right) - \mu^{2} \\
& = \sum_{i=1}^n w_i(\sigma_i^2 + \mu_i^2)- \mu^2 & ( \sigma_i^2 = \operatorname{E}[X_i^2] - \mu_i^2 \implies \operatorname{E}[X_i^2] = \sigma_i^2 + \mu_i^2) 
\end{align}</math>

These relations highlight the potential of mixture distributions to display non-trivial higher-order moments such as [[skewness]] and [[kurtosis]] ([[fat tail]]s) and multi-modality, even in the absence of such features within the components themselves.  Marron and Wand (1992) give an illustrative account of the flexibility of this framework.<ref name="Marron92">{{Cite journal|title=Exact Mean Integrated Squared Error |first1=J. S. |last1=Marron |first2=M. P. | last2=Wand | journal=[[The Annals of Statistics]]|volume=20 |year=1992| pages=712–736 |issue=2 | doi=10.1214/aos/1176348653|doi-access=free }}, http://projecteuclid.org/euclid.aos/1176348653</ref>