Content deleted Content added
KGrammatikos (talk | contribs) m →Probability distribution: Corrected error on the variance of exponential families, verified with reference 2. |
KGrammatikos (talk | contribs) m →Probability distribution: Language revision that allows readers to understand the conditions under which the equations written below are true. Crosschecked expressions with ref. 2 in the article |
||
Line 48:
: <math> f_Y(y \mid \theta, \tau) = h(y,\tau) \exp \left(\frac{b(\theta)T(y) - A(\theta)}{d(\tau)} \right). \,\!</math>
<math>\boldsymbol\theta</math> is related to the mean of the distribution. If <math>\mathbf{b}(\boldsymbol\theta)</math> is the identity function, then the distribution is said to be in [[canonical form]] (or ''natural form''). Note that any distribution can be converted to canonical form by rewriting <math>\boldsymbol\theta</math> as <math>\boldsymbol\theta'</math> and then applying the transformation <math>\boldsymbol\theta = \mathbf{b}(\boldsymbol\theta')</math>. It is always possible to convert <math>A(\boldsymbol\theta)</math> in terms of the new parametrization, even if <math>\mathbf{b}(\boldsymbol\theta')</math> is not a [[one-to-one function]]; see comments in the page on [[exponential families]]. If, in addition, <math>\mathbf{T}(\mathbf{y})</math>
:<math> \boldsymbol\mu = \operatorname{E}(\mathbf{y}) = \
For scalar <math>\mathbf{y}</math> and <math>\boldsymbol\theta</math>, this reduces to
Line 55:
Under this scenario, the variance of the distribution can be shown to be<ref>{{harvnb|McCullagh|Nelder|1989}}, Chapter 2.</ref>
:<math>\operatorname{Var}(\mathbf{y}) = \nabla^
For scalar <math>\mathbf{y}</math> and <math>\boldsymbol\theta</math>, this reduces to
| |||