# Identically distributed?

Given a set of independent and identically distributed random variables $$(\xi_i)_{i\in I}$$ i know that $$(X_i)_{i\in I}$$ and $$(Y_i)_{i\in I}$$ ar conditionally independent and identically distributed. And $$(X_i)_{i\in I}$$ is a family of identically distributed random variables.

So I want to show, that $$(X_i, Y_i)_{i\in I}$$ are identically distributed. So let $$\mathcal{F}:=\sigma( \xi_i: i\in I)$$ then we have for $$\Delta \in I$$ and $$f,h$$ bounded and continous:

$$E\left[ f(X_\mathbf{i}) \cdot h(Y_\mathbf{i}) \right]\\ =E\left[ E\left[ f(X_\mathbf{i}) \cdot h(Y_\mathbf{i}) | \mathcal{F} \right] \right]\\ =E\left[ E\left[ f(X_\mathbf{i}) | \mathcal{F} \right] E\left[ h(Y_\mathbf{i}) | \mathcal{F}_{\mathbf{i} } \right] \right]\\ =E\left[ E\left[ f(X_\mathbf{i}) | \mathcal{F} \right] E\left[ h(X_\mathbf{i}) | \mathcal{F} \right] \right]\\ \overset{ X_\mathbf{i}\overset{(d)}{=}X_\Delta }{=}E\left[ E\left[ f(X_\Delta) | \mathcal{F} \right] E\left[ h(X_\Delta) | \mathcal{F} \right] \right]\\ =E\left[ E\left[ f(X_\Delta) | \mathcal{F} \right] E\left[ h(Y_\Delta) | \mathcal{F} \right] \right]\\ =E\left[ E\left[ f(X_\mathbf{i}) \cdot h(Y_\Delta ) | \mathcal{F} \right] \right]\\ = E\left[ f(X_\Delta) \cdot h(Y_\Delta) \right]$$

So that is true for every $$f,h$$ bounded and continous. That means $$(X_i, Y_i)\overset{(d)}{=} (X_j, Y_j)$$ for $$i,j\in i$$.

Is t really so easy or am I missing something important here?

Thx...