Initial Probability Vector

Topics: Stochastic Process

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(definition)

Given a stochastic process with $m$ possible states, we can define an initial probability vector $a$:

$$a=(a_0,a_1,\dots ,a_m)$$

…where $a_i$ denotes the probability of the initial state being $s_i$.

Example

For instance, let’s say we have the stochastic process of tossing a coin and noting when we get heads or when we get tails. This process has a simple state set of $S=\left(\text{heads},\text{tails}\right)$.

The initial probability vector of this process is $a=\left(\frac{1}{2},\frac{1}{2}\right)$, since the probability of getting either heads or tails is $\frac{1}{2}$.