# Trouble Understanding the Integral Underlying Random Utility Models

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.

• Did you mean $\forall j \ne i$? – Herr K. Apr 14 at 17:09
• Yes I did, thank you – jl_1995_95 Apr 14 at 17:12

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.