# How to express this integral in the Mathematica?

dear All!

I'm a beginner in Mathematica coding, and now trying to find out the following (numerical) integration using Mathematica: \begin{align} g(\mu) := \int_0^1 \sum_{i=1}^{K} [x f(0.05+0.95x; \mu_i) \prod_{j \neq i} F(x; \mu_j)] dx \end{align} where $$\mu = (\mu_1, ... , \mu_K)$$ is a $$k$$-vector, and $$F(x; \theta), f(x; \theta)$$ are given functions, i.e. \begin{align} & F(x; \theta) = \Phi[\theta - \Phi^{-1}(1 - \frac{x}{2})]; \\ & f(x; \theta) = e^{-\frac{\theta^2}{2}} \cosh(\theta \Phi^{-1}(1 - \frac{x}{2})) \end{align} The Mathematica code for the two functions are:

F[x_, \[Theta]_] := 1 - CDF[NormalDistribution[0, 1], InverseCDF[NormalDistribution[0, 1],  1 - x/2] - \[Theta]] + CDF[NormalDistribution[0,1], -InverseCDF[NormalDistribution[0, 1],  1 - x/2] - \[Theta]]
f[x_, \[Theta]_] := Exp[-\[Theta]^2/2] * Cosh[\[Theta] * InverseCDF[NormalDistribution[0, 1], 1 - x/2]]


I know how to product all elements in the list using Times, but what is the summation of production like in the integral? Thanks so much!!

k = 5;