Binary Hypothesis

Given a set of data

$$\mathbf{x}={\left[\begin{array}{cccc}x(0)& x(1)& \dots & x(N-1)\end{array}\right]}^T$$

Where the Observation Space, $\Omega$, is given such that $x\in \Omega$.

We then have to decide which distribution the data may have been sampled. We want to observe the validity of the following two hypotheses

$$\begin{array}{rl}\text{Null Hypothesis}& H_0:x\sim {\mathcal{P}}_0\\ \text{Alternative}& H_1:x\sim {\mathcal{P}}_1\end{array}$$

Where ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$ are the distributions we are testing for.

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Note: Changing the threshold changes all the probabilities.