# Posterior Probability Function

## References

### 2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Posterior_probability
• The posterior probability of a Random Event or an uncertain proposition is the Conditional Probability that is assigned after the relevant evidence is taken into account.
• The posterior probability distribution of one Random Variable given the value of another can be calculated with Bayes' Theorem by multiplying the Prior Probability Distribution by the Likelihood Function, and then dividing by the Normalizing Constant, as follows:
• $\displaystyle{ f_{X\mid Y=y}(x)={f_X(x) L_{X\mid Y=y}(x) \over {\int_{-\infty}^\infty f_X(x) L_{X\mid Y=y}(x)\,dx}} }$
• gives the posterior Probability Density Function for a random variable $\displaystyle{ X }$ given the data $\displaystyle{ Y }$ = $\displaystyle{ y }$, where
• $\displaystyle{ f_X(x) }$ is the prior density of X,
• $\displaystyle{ L_{X\mid Y=y}(x) = f_{Y\mid X=x}(y) }$ is the likelihood function as a function of x,
• $\displaystyle{ \int_{-\infty}^\infty f_X(x) L_{X\mid Y=y}(x)\,dx }$ is the normalizing constant, and
• $\displaystyle{ f_{X\mid Y=y}(x) }$ is the posterior density of $\displaystyle{ X }$ given the data $\displaystyle{ Y }$ = y.
• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Fair_coin
• In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin.