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PROBABILITY

DEFINITION

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Classic Probability $$P=\frac{#A}{#S}$$

RELATIONSHIP

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Conditioned Probability $$P(A\vert B)=P_B(A)= \frac{P(A\cap B)}{P(B)}$$
Bayer Probability Formula $$P(A\vert B)= P(B\vert A)\cdot \frac{P(A)}{P(B)}$$
Independent Event $$P(A\vert B)=P(A) \Leftrightarrow \frac{P(A\cap B)}{P(B)}=P(A)$$
Total Probability Formula $$P(A)=\sum_{i=1}^{N}P(A\vert B_i)\cdot P(B_i)$$
Composite Probability Formula $$P\left(\bigcap^N_{i=1} R_i\right)=\left[\prod^N_{q=2} P\left(R_q\vert\bigcap_{m=1}^{q-1}R_m\right)\right]\cdot P(R_1)$$
@De Morgan Rules $$\left(\bigcap_i^{n \vert \infty}A_i\right)^C=\bigcup_i^{n \vert \infty} A_i^C$$ $$\left(\bigcup_i^{n \vert \infty}A_i\right)^C=\bigcap_i^{n \vert \infty} A_i^C$$

SYSTEM RELIABILITY

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Parallel Connection $$P(S)=P\left(\bigcup_i S_i\right)=1-\prod_i(1-P(S_i))$$
Series Connection $$P(S)=P\left(\bigcap_i S_i\right)=\prod_i P(S_i)$$

COMBINATION

PERMUTATION

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Simple Permutation $$n!$$
Permutation with Repetition $$\frac{n!}{\prod x!}$$

COMBINATION

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Simple Combination $$\left( \begin{array}{cc} n \ k\end{array}\right)$$

DISPOSITION

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Disposition with Repetition $$n^k$$
Simple Disposition $$\frac{n!}{(n-k)!}$$

RANDOM VARIABLE

DISCRETE RANDOM VARIABLE

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@Discrete Random Variable $$P(x_\alpha)= F(x_\alpha)-F(x_{\alpha-1})$$

BERNULLI RANDOM VARIABLE

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Expected Value $$E(X)=p$$
@Variance $$Var(X)= p-p^2$$

BINOMIAL RANDOM VARIABLE

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Probability $$P(X=\alpha)=\left(\begin{array}{cc} n\ k \end{array}\right)\cdot p^k\cdot(1-p)^{n-k}$$
Expected Value $$E(X)=n\cdot p$$
@Variance $$Var(X)= n\cdot(p-p^2)$$

GEOMETRIC RANDOM VARIABLE

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Probability $$P(X=\alpha)=(1-p)^{\alpha-1}\cdot p$$
Expected Value $$E(X)=\frac{1}{p}$$
@Variance $$Var(X)= \frac{1-p}{p^2}$$

POISSON RANDOM VARIABLE

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Probability $$P(X=\alpha)=\frac{\lambda^\alpha}{\alpha!}\cdot e^{-\lambda}$$
Expected Value $$E(X)=\lambda$$
@Variance $$Var(X)=\lambda$$

INDEX

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⭐ Expected Value $$\mu_x=E(X)=\begin{cases}\sum_{S_X} x_k\cdot p(x_k) &\text{ IF Discrete}\ \int_{S_X} x_k\cdot f(x_k) &\text{ IF Continuous}\end{cases}$$
Covariance $$Cov(X_1,\dots X_n)=\sigma(X_1,\dots X_n)=\sigma_{X_1,\dots X_n}=E\left(\left[\prod_i X_i-\mu_{X_i} \right]^2\right)=E\left(\prod_i X_i\right)-\prod_i \mu_{X_i}$$ $$Cov(X,Y)=\begin{cases} \sum_{x_i}\sum_{y_j} (x_i-\mu_x)\cdot(y_j-\mu_y)\cdot \mathbb{P}(X,Y)\cdot(x_i,y_j) &\text{ IF Discrete}\ \int_{S_X}\int_{S_Y} (x-\mu_x)\cdot(y-\mu_y)\cdot f(X,Y)\cdot(x,y)\cdot dx\cdot dy &\text{ IF Continuous}\end{cases} $$
@Variance $$\sigma_X^2=Var(X)=E([X-\mu_x]^2)=E(X^2)-\mu_x^2=\begin{cases} \sum_{S_X} (x_k-\mu_X)^2\cdot p(x_k) &\text{ IF Discrete}\ \int_{S_X} (x_k-\mu_X)^2\cdot f(x_k) &\text{ IF Continuous}\end{cases}$$
⭐ Standard Deviation $$\sigma_X=\sqrt{Var(X)}$$

RANDOM VARIABLE VECTOR

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⭐ Linear Correlation Index $$\rho(X,Y)= \frac{Cov(X,Y)}{\sigma_X\cdot\sigma_Y} = E\left( \frac{X-\mu_x}{\sigma_X} \cdot \frac{Y-\mu_Y}{\sigma_Y}\right)$$

CARDINALITY

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$$#(A\cup B)=#A+#B-#(A\cap B)$$
$$#\left(\bigcup_i^n A_i\right)=\sum_i #A_i-\sum_i\sum_j#(A_i\cap A_j)+#\bigcap_i^n A_i$$

RANDOM VARIABLE

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Survival Function $$R_X(x)=1-F_X(x)$$

PASCAL RANDOM VARIABLE

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Probability $$P(X=\alpha)=\left(\begin{array}{cc} \alpha-1\ \beta\end{array}\right)\cdot p^\beta\cdot(1-p)^{\alpha-\beta}$$

HYPERBOLIC RANDOM VARIABLE

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Probability $$P(X=\alpha)= \frac{ \left(\begin{array}{cc} r\ \alpha\end{array}\right) \cdot \left(\begin{array}{cc} b\ n-\alpha\end{array}\right)}{\left(\begin{array}{cc} N\ n\end{array}\right)}$$
Expected Value $$E(X)=n\cdot \frac{r}{N}$$
@Variance $$Var(X)= E(X)\cdot\left(1- \frac{r}{N}\right)\cdot \frac{N-n}{N-1}$$

CONTINUOUS RANDOM VARIABLE

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@Continuous Random Variable $$$$

NORMAL RANDOM VARIABLE

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Probability $$P(X=\alpha)= \frac{1}{\sqrt{2\pi}\cdot\sigma}\cdot e^{- \frac{1}{2}\cdot(\frac{x-\mu}{\sigma})^2}$$
Expected Value $$E(X)=\mu$$
@Variance $$Var(X)= \sigma^2$$

STANDARD NORMAL RANDOM VARIABLE

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Probability $$P(X=\alpha)= \frac{1}{\sqrt{2\pi}}\cdot e^{- \frac{x^2}{2}}$$
Expected Value $$E(X)=\mu=0$$
@Variance $$Var(X)= \sigma^2=1$$

EXPONENTIAL RANDOM VARIABLE

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Probability $$f_X(\alpha)=\lambda\cdot e^{-\lambda\cdot x}\cdot\mathbb{1}$$
Expected Value $$E(X)=\frac{1}{\lambda}$$
@Variance $$Var(X)=\frac{1}{\lambda^2}$$
Distribution Function $$F_X(x)=\begin{cases} 0 &\text{IF } x<0\ 1-e^{-x\cdot\alpha} &\text{IF } x \geqslant0\end{cases}$$
Hazard Rate $$\tau(t)=\lambda$$

WEIBULL RANDOM VARIABLE

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Probability $$f_X(\alpha)=\lambda\cdot e^{-\lambda\cdot x}\cdot\mathbb{1}$$
Hazard Rate $$\tau(t)=\frac{b}{a^b}\cdot t^{b-1}$$

UNIFORM RANDOM VARIABLE

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Probability $$f_X(x)=\begin{cases} \frac{1}{b-a} &\text{IF }a\leqslant x\leqslant b\ 0 &\text{ELSE}\end{cases}$$
Expected Value $$E(X)=\frac{a+b}{2}$$
@Variance $$Var(X)=\frac{(b-a)^2}{12}$$

INDEX

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n-th Moment $$E(X^n)=\begin{cases} \sum_{S_k} x_k^n\cdot p(x_k) &\text{ IF Discrete}\ \int_{S_k} x_k^n\cdot f(x_k) &\text{ IF Continuous}\end{cases}$$
Covariance $$Cov(X_1,\dots X_n)=E\left(\prod_i X_i\right)-\prod_i \mu_{X_i}=Cov(X,Y)=\begin{cases} \sum_{x_i}\sum_{y_j} (x_i-\mu_x)\cdot(y_j-\mu_y)\cdot \mathbb{P}(X,Y)\cdot(x_i,y_j) &\text{ IF Discrete}\ \int_{S_X}\int_{S_Y} (x-\mu_x)\cdot(y-\mu_y)\cdot f(X,Y)\cdot(x,y)\cdot dx\cdot dy &\text{ IF Continuous}\end{cases}$$
⭐ @Linear Correlation Index $$\rho(X,Y)= \frac{Cov(X,Y)}{\sigma_X\cdot\sigma_Y} = E\left( \frac{X-\mu_x}{\sigma_X} \cdot \frac{Y-\mu_Y}{\sigma_Y}\right)$$

VECTOR RANDOM VARIABLE

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Convolution $$P_Z=P_X ;^*; P_Y=\begin{cases} \sum_{x\epsilon S_X} p_X(x)\cdot p_Y(z-x) &\text{IF Discrete}\ \int_{S_X} f_X(x)\cdot f_Y(z-x)\cdot dx &\text{IF Continuous} \end{cases}$$