# What is the probability/likelihood of a sample being drawn from a probability distribution over binary values

Suppose we have a known discrete probability distribution $$X$$ over $$\{0,1\}^k$$.
Given a sequence of binary values $$e = (e_1, ..., e_n)\text{, where } e_i\in \{0,1\}^k$$, what is the probability (or the likelihood) that $$e$$ was drawn from the Distribution $$X$$?
Is there for example a bound that tells us that such a sequence of samples $$e$$ is very unlikely to be drawn from $$X$$, and can thus be ignored?

• So you're asking for the density function of all binary sequences of length $n$? Seems like they would all be equiprobable from how you have described the problem. Are you thinking of a particular type of sequences (maybe sequences which have some sort of autocorrelation structure)? – Demetri Pananos Nov 13 '18 at 6:14
• @DemetriPananos Actually X is a noise distribution which is not uniform. So the values of $e_i$ are drawn non uniformally. X would have maybe some distribution function, but I'm aiming more at a general case. – abdul rahman taleb Nov 13 '18 at 7:29

At the moment you have not imposed any conditions on the distribution other than saying that it is a distribution on the set of binary vectors of length $$k$$. Your distribution can therefore be expressed as a probability vector over the $$2^k$$ possible states of the random variable. If the distribution is known, as you state, then presumably you can see exactly which outcomes are very unlikely.