Multivariate and integral representations explanation

Question

I would to understand more on the multivariate and integral representations. I am fascinated by the possibility of a multivariate distribution like this one:

$$\mathbb{P}(X_1 \le Z + a, X_2 \le Z + a, ..., X_n \le Z + a),$$

to be transformed into:

$$\int \limits_{- \infty} ^{+ \infty} {\Phi(z+a)^n \phi(z)dz}.$$

and vice-versa. Here $\Phi(z)$ and $\phi(z)$ are, respectively, the cdf and the pdf. Everything started reading about this: Integrating pdf times cdf squared.

How is that a random vector is transformed into a scalar? is this some sort of projection? is there an analog transformation between a euclidean vector and a scalar?

Answer 1

Lets assume $Z$ is a random variable defined over a support of $\mathbb{R}$ and $X_1,...,X_n$ are i.i.d and have cdf $\Phi$, in which case

\begin{align} \text{Prob}(a) &\equiv \mathbb{P}(X_1 \le Z + a, X_2 \le Z + a, ..., X_n \le Z + a) \\[18pt] &= \mathbb{E}_Z[\mathbb{P}(X_1 \le Z + a, X_2 \le Z+ a, ..., X_n \le Z + a | Z = z)] \\[12pt] &= \int \limits_{-\infty}^{\infty} \mathbb{P}(X_1 \le Z + a, X_2 \le Z+ a, ..., X_n \le Z + a | Z = z) \phi(z) \,dz \\[6pt] &= \int \limits_{-\infty}^{\infty}\mathbb{P}(X_1 \le z + a, X_2 \le z + a, ..., X_n \le z + a) \phi(z) \,dz \\[6pt] &= \int \limits_{-\infty}^{\infty}\mathbb{P}(X_1 \le z + a) \cdot \mathbb{P}(X_2 \le z + a) \cdots \mathbb{P}(X_n \le z + a) \phi(z) \,dz \\[6pt] &= \int \limits_{-\infty}^{\infty}\Phi(z + a)^n \phi(z) \,dz. \\[6pt] \end{align}

Intuitively, $z$ (the variable of integration) is a scalar as opposed to a vector because it is the same variable that all $X_1,...,X_n$ are being compared with, and it allows us to compute the total probability utilizing the conditional probability given $Z=z$, which is much easier probability to compute. This alternative, easier formulation is allowed as per the law of total probability.

Observe that the integration variable would instead be the vector $(z_1,...,z_n$) if we were instead computing $\mathbb{P}(X_1 \le Z_1 + a, X_2 \le Z_2 + a, ..., X_n \le Z_n + a.)$