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I was reading through the textbook of Introduction to Machine Learning, it introduces $z^{t} = (z_{1}^{t},...,z_{k}^{t})$, where $z_{i}$ is an indicator variable, each with probability $\pi_{i}$, then the textbook gives the expression for the pmf for z: $p(z) = \prod_{i}^{k}\pi_{i}^{z_{i}^{t}}$.

I would like to know the detailed derivation of this formula, my concern is that according to this formula below, there should be another term in front of the product term. enter image description here

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If you have a sequence of Bernoulli (indicator) random variables $Z_1,Z_2,Z_3,...$ with respective probabilities $\pi_1, \pi_2, \pi_3, ...$ then for the sample vector $\mathbf{z}_k = (z_1,...,z_k)$ the probability mass function should be:

$$\begin{equation} \begin{aligned} p_\mathbf{Z}(\mathbf{z}_k | \boldsymbol{\pi}) \equiv \mathbb{P}(\mathbf{Z}_k = \mathbf{z}_k | \boldsymbol{\pi}) &= \prod_{i=1}^k \mathbb{P}(Z_i = z_i | \pi_i) \\[6pt] &= \prod_{i=1}^k \text{Bern}(z_i | \pi_i) \\[6pt] &= \prod_{i=1}^k \pi_i^{z_i} (1-\pi_i)^{1-z_i}. \\[6pt] \end{aligned} \end{equation}$$

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