Visions of Mathematics: Probability

$P(A\cup B) = P(A) \: + \: P(B) - P(A\cap B)$

$P(A\cap B) = P(A) \: + \: P(B|A)$

$P(A| B) = \frac{P(B|A) P(A)}{P(B|A) P(A) + P(B|A') P(A')}$

Discrete Distributions
For a random varibale X taking values of $x_i$ with probabilities $P(X=x_i)$

Mean, $\mu: \mathbb{E}[x] = \sum x_i \: * \: P(X=x_i)$

Variance , $\sigma^2: \: \: Var[x] = \sum (x_i - \mu)^2 * P(X=x_i) = \sum x_i^2 * P(X=x_i) - \mu^2$

Binomial Distribution
$\\*P(X=x)= \binom{n}{x} p^x (1-p)^{n-x}; \\*\: \: \textbf{mean }=\mu = np;\: \: \textbf{variance} = \: \: \sigma^2 = np(1-p)$

$Image result for binomial distribution$

Possion Distribution
$\\*P(X=x)= e^{-\lambda} \frac{\lambda^x}{x!} \\*\: \: \textbf{mean }=\mu = \lambda;\: \: \textbf{variance} = \: \: \sigma^2 = \lambda$

Geometric Distribution
$\\*P(X=x)= p (1-p)^{n-1} \\*\: \: \textbf{mean }=\mu = \frac{1}{p} ;\: \: \textbf{variance} = \: \: \frac{1-p}{p^2}$

Uniform Distribution
$\\*P(X=x)= \frac{1}{n} \\*\: \: \textbf{mean }=\mu = \frac{n+1}{2}\: \: \textbf{variance} = \: \: \frac{n^2-1}{12}$

Erlang Distribution
$\\*P(X=x)= \frac{\lambda e^{-\lambda x} (\lambda x)^{r-1}}{(r-1)!} \\*\\*\: \: \textbf{mean }=\mu = \frac{r}{\lambda}\: \: \textbf{variance} = \: \: \frac{r}{\lambda^2}$

Gamma Distribution

$\\*P(X=x)= \frac{\lambda e^{-\lambda x} (\lambda x)^{\alpha-1}}{\Gamma \alpha} \\*\\*\: \: \textbf{mean }=\mu = \frac{\alpha}{\lambda};\: \: \textbf{variance} = \: \: \frac{\alpha}{\lambda^2}$

Exponential Distribution

$\\*P(X=x)= \lambda e^{-\lambda x} \\*\\*\: \: \textbf{mean }=\mu = \frac{1}{\lambda};\: \: \textbf{variance} = \: \: \frac{1}{\lambda^2}$

Normal Distribution

$\\*P(X=x)= \frac{1}{\sigma \sqrt{2 \pi}}e^{-[\frac{(x-\mu)^2}{2\sigma^2}]} \\*\\*\: \: \textbf{mean }=\mu;\: \: \textbf{variance} = \: \: \sigma^2$