The Markov Property

For a sequence of Random Variables $Z_1,Z_2,...,Z_n$ where each random variable is independent, we can write their joint distribution as follows

$f(z_1,z_2,...,z_n)=f(z_1)f(z_2)...f(z_n)$ However, if the sequence of Random Variables is completely dependent, we will need to write the conditional probability of each $Z_i$ ad a condition probability of all other $Z_j$ such that $i\ne j$ and $j<i$ for some fixed $i$ and $j\in \mathrm{1...}i-1$ .

$$f(z_1,z_2,\dots ,z_n)=f(z_n\mid z_1,z_2,\dots ,z_{n-1})f(z_{n-1}\mid z_1,\dots ,z_{n-2})\cdots f(z_4\mid z_1,z_2,z_3)f(z_3\mid z_1,z_2)f(z_2\mid z_1)f(z_1).$$

What if each random variable is only dependent on the previous random variable in the sequence? This is the Markov Property.

Markov Property

$$f(z_1,z_2,\dots ,z_n)=f(z_n\mid z_{n-1})f(z_{n-1}\mid z_{n-2})\cdots f(z_4\mid z_3)f(z_3\mid z_2)f(z_2\mid z_1)f(z_1).$$