PROBABILITIES Course

Summary

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A summary of the discrete distributions

Name Desc PMF $\mathbb{P}(X=k)$ $\mathbb{E}$ $\mathbb{V}$ CDF MGF
Bernoulli $B(p)$ Distribution for binary variables. $p^k * (1-p)^{1-k}$ $p$ $p * (1-p)$ $1-p$ $(1-p)+p*e^t$
Binomial $B(n, p)$ A repetition of $n$ Bernoulli distributions. $ C_n^k * p^k * (1-p)^{n-k}$ $n*p$ $n * p * (1-p)$ messy $((1-p)+p*e^t)^n$
Discrete uniform distribution $U([a,b])$ Every outcome has the same probability. $\frac{1}{b-a+1}$ $\frac{a+b}{2}$ $\frac{(b-a)(b-a+2)}{12}$ $\frac{\lfloor k\rfloor-a+1}{b-a+1}$ messy
Discrete uniform distribution $U([1,n])$ Every outcome has the same probability. $\frac{1}{n}$ $\frac{n+1}{2}$ $\frac{n^2 - 1}{12}$ $\frac{\lfloor k\rfloor}{n}$ messy
Geometric $G(p)$ The probability of $k$ being a success after $k-1$ failures with a probability $p$. $(1-p)^{k-1} * p$ $\frac{1}{p}$ $\frac{1-p}{p^2}$ $1-(1-p)^k$ $\frac{p * e^t}{1-(1-p) * e^t}$
Hypergeometric $H(N, k, n)$ Distribution without replace of $n$ trials with $K$ out of $N$ elements. ${{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}$ $\frac{K}{N}$ $\mathbb{E}(X) * (1 - \frac{K}{N}) * \frac{N-n}{N-1}$ messy messy
Poisson $\mathbb{P}(\lambda)$ A big $n$ and a small probability $p$ giving us $\lambda = n*p$. $\frac{\lambda^k * e^{-\lambda}}{k!}$ $\lambda$ $\lambda$ messy $e^{\lambda * (e^{t}-1)}$

and the continuous distributions

Name Desc PDF $\mathbb{E}$ $\mathbb{V}$ CDF MGF
Continuous uniform distribution $U([a,b])$ Every outcome has the same probability. $f_X(x) = \frac{1}{b-a}$ $\frac{a+b}{2}$ $\frac{(b-a)^2}{12}$ messy messy
Exponential distribution $E(\lambda)$ ??? $f_X(x) = \lambda e^{-\lambda{x}}$ $\frac{1}{\lambda}$ $\frac{1}{\lambda^2}$ $1-e^{-\lambda x}$ $\frac{\lambda}{\lambda-t}$
Normal/Gaussian distribution $N(\mu, \sigma^2)$ mu ($\mu$) is the mean and sigma ($\sigma$) is the standard deviation. $f_X(x) = {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}$ $\mu$ $\sigma^2$ $messy$ $e^{\mu t+ (\sigma^{2} * t^{2})/2}$
Standard Normal distribution $N(0, 1)$ This is a particular case of normal distribution. $f_X(x) = {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}$ $0$ $1$ $messy$ $e^{\mu t+ (\sigma^{2} * t^{2})/2}$