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\title{SEE: Detail of Purchase Decision calculation}
\author{Willy H. Gerber}

\begin{document}

\title{Detail of Purchase Decision calculation}

Probability of purchasing a product of wellbeing $ \omega $

\begin{equation}
p_{i} = \dfrac{1}{e^{-\beta\omega_{i} + \alpha} + 1}
\end{equation}

The welleing is defined as

\begin{equation}
\omega_{i} = \varepsilon_{i} + \vec{\mu}\cdot\vec{s_{i}}
\end{equation}

$ \alpha $ must be choisen to fulfill:

\begin{equation}
\sum_{i} p_{i} = 1
\end{equation}

\begin{equation}
e^{-\beta\omega_{i} + \alpha} + 1 = \dfrac{1}{p_{i}}
\end{equation}

\begin{equation}
e^{-\beta\omega_{i} + \alpha} = \dfrac{1}{p_{i}} - 1
\end{equation}

\begin{equation}
e^{-\beta\omega_{i} + \alpha} = \dfrac{1 - p_{i}}{p_{i}}
\end{equation}

\begin{equation}
-\beta\omega_{i} + \alpha = \ln\left(\dfrac{1 - p_{i}}{p_{i}}\right)  \equiv z_{i}
\end{equation}

\begin{equation}
z_{i} = -\beta\varepsilon_{i} -\beta\vec{\mu}\cdot\vec{s_{i}} + \alpha
\end{equation}

\begin{equation}
(\varepsilon,\vec{s})
\end{equation}


\end{document}