Fitting Gaussian

Overview


Hardin Formulation


{% \vec{y} = x \vec{\beta} + \vec{\epsilon} %}
The probabilty density is
{% f(y;\mu,\sigma) = \frac{1}{\sqrt{2 \pi \sigma^2}} exp [- \frac{(y-\mu)^2}{2\sigma^2}] %}
(see normal distribution)
{% f(y;\mu,\sigma) = exp[- \frac{(y-\mu)^2}{2 \sigma ^2} - \frac{1}{2} ln(2 \pi \sigma ^2)] %}
{% = exp[\frac{y \mu - \mu^2/2}{\sigma ^2} - \frac{y^2}{2 \sigma ^2} - \frac{1}{2}ln(2 \pi \sigma^2)] %}
{% f(y; \theta, \phi) = exp[\frac{y \theta - b(\theta)}{a(\phi)} + c(y,\phi)] %}
  • {% \theta = \mu %}
  • {% b(\theta) = \frac{\mu ^2}{2} %}

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