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• O = {o1, …, on} (n observed objects), Θ = {θ1, …, θk} (parameters of the k distributions), and Pj(oi| θj) is the probability that oi is generated from the j-th distribution using parameter θj, we have

\[P(o_{i}|\theta)=\sum_{j=1}^{k}w_{j}P_{j}(o_{i}|\theta_{j} )\]

\[P(O|\theta)=\prod_{i=1}^{n}\sum_{j=1}^{k}w_{j}P_{j}(o_{i}|\theta_{j} )\]

• Univariate Gausian mixture model
• Assume the probability density function of each cluster follows a 1-d Gaussian distribution. Suppose that there are k clusters.
• The probability density function of each cluster are centered at μj with standard deviation σj, θj, = (μj, σj), we have

\[P(o_{i}|\theta_{j})=\frac{1}{\sqrt{2\pi}\sigma_{j} }e^{-\frac{(o_{i}-\mu_{j})^2 }{2\sigma^{2} }}\]

\[P(o_{i}|\theta)=\sum_{j=1}^{k}\frac{1}{\sqrt{2\pi}\sigma_{j} }e^{-\frac{(o_{i}-\mu_{j})^2 }{2\sigma^{2} }}\]

\[P(O|\theta)=\prod_{i=1}^{n}\sum_{j=1}^{k}\frac{1}{\sqrt{2\pi}\sigma_{j} }e^{-\frac{(o_{i}-\mu_{j})^2 }{2\sigma^{2} }}\]