3 Using Equations

Example of equations

\[\begin{equation} Y \sim \textrm{Bernoulli}(\pi) \tag{3.1} \end{equation}\]

\[\begin{equation} \pi = P(Y=1 \vert x; \theta) = F(x; \theta) \tag{3.2} \end{equation}\]

\[\begin{equation} P(Y=y | x; \theta) = F(x;\theta)^y(1-F(x;\theta))^{1-y} \tag{3.3} \end{equation}\]

The likelihood function \(\mathcal{L}\) can be determined using equation (3.3)

\[\begin{equation} \begin{split} \mathcal{L}(\theta | y, x) &= \prod_{i}^{N} P(y_i | x_i; \theta) \\ &= \prod_{i}^{N}F(x_i;\theta)^{y_i}(1-F(x_i;\theta))^{1-y_i} \end{split} \tag{3.4} \end{equation}\]

Equation (3.4) is commonly expressed in terms of its logarithm.

\[\begin{equation} \ln \mathcal{L}(\theta | y, x) = \sum_{i}^{N} y_i \ln\left(F(x_i;\theta)\right) + (1-y_i) \ln\left(F(x_i;\theta))\right) \tag{3.5} \end{equation}\]