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Given a utility function, $U_{nj} = V_{nj} + \varepsilon_{nj}$, it makes sense that we can find the probability the decision maker $n$ chooses alternative $i$ as:

$$Pr(U_{ni} > U_{nj} \forall j \neq i)$$ $$=Pr(V_{ni} + \varepsilon_{ni} > V_{nj} + \varepsilon_{nj} \forall j \neq i)$$ $$=Pr(\varepsilon_{nj} - \varepsilon_{ni} < V_{ni} - V_{nj} \forall j \neq i)$$

I am having an issue understanding the next step though, where:

$$Pr(\varepsilon_{nj} - \varepsilon_{ni} < V_{ni} - V_{nj} \forall j \neq i)$$ $$= \int_{\varepsilon} \text{I}(\varepsilon_{nj} - \varepsilon_{ni} < V_{ni} - V_{nj} \forall j \neq i)f(\varepsilon_n)d\varepsilon_n$$

Specifically, I am having trouble wrapping my head around why the joint density function, $f(\varepsilon_n)$, comes into play. I understand that this is one of the crucial aspects to understanding discrete choice models, since the choice of this joint density function underlies what model is to be used. Hence, I would like to gain a better understanding of why it is in here. If anyone would maybe have an example or a dumbed-down explanation of why this integral is what it is, I would appreciate it.

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    $\begingroup$ Did you mean $\forall j \ne i$? $\endgroup$ – Herr K. Apr 14 at 17:09
  • $\begingroup$ Yes I did, thank you $\endgroup$ – jl_1995_95 Apr 14 at 17:12
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This has nothing to do with any specific model. For any event $A$, let $I_A$ be the indicator function such that $I_A(\omega)=1$ if $\omega\in A$ and $I_A(\omega)=0$ if $\omega\notin A.$

Then $\mathbb{P}(A)=\mathbb{E}[I_A],$ and here the expectation is given in terms of a density function.

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