Discrete Distribution¶
Binomial Distribution¶
\[Pr(X=x)=\binom{n}{x}p^x{(1-p)}^{n-x}\]
Bernoulli Distribution¶
A Bernoulli distribution is a special case of binomial distribution. Specifically, when n=1, i.e. x only could be 0 or 1, the binomial distribution becomes Bernoulli distribution.
Geometric Distribution¶
The geometric distribution gives the probability that the first occurrence of success requires k independent trials, each with success probability p. If the probability of success on each trial is p, then the probability that the kth trial (out of k trials) is the first success is
\[Pr(X=x)=(1-p)^{x-1}p\]
Poisson Distribution¶
Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time.
\[\begin{split}Pr(X=x)=\frac{\lambda^x e^{-\lambda}}{x!}, \\
\lambda = E(X) = Var(X)\end{split}\]