Consider a random vector $$ X\equiv \begin{pmatrix} X_1\\ X_2\\ X_3 \end{pmatrix} $$ with pdf $$f(x)=\overbrace{\sum_{k=1}^ K \frac{1}{K} f_k(x)}^{\text{finite mixture}}$$ and $\forall k=1,...,K$ $f_k(x)$ is the pdf of a 3-variate Normal with mean $\begin{pmatrix} \mu_k\\ \mu_k\\ \mu_k \end{pmatrix}$ and variance-covariance matrix $$ \Sigma_k\equiv \begin{pmatrix} a_k & b_k & b_k\\ b_k & a_k & b_k\\ b_k & b_k & a_k \end{pmatrix} $$

Question: are the random variables $\{X_1, X_2, X_3\}$ exchangeable?

My intuition is that the answer is yes, because every component of the mixture induces exchangeability. However, I would like to hare your opinion.

  • 3
    $\begingroup$ Since $f(x_1,x_2,x_3)=f(x_2,x_3,x_1)=...$ the components of $X$ are indeed exchangeable. $\endgroup$ – Xi'an Feb 5 at 21:17

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