# Probability - Probability formulas

Events : A, B
Probability : P
Random variables : X, Y , Z
Values of random variables : x,y,z
Expected value of X : $\mu$
Any positive real number  : $\varepsilon$
Standard deviation : $\sigma$
Variance : $\sigma^2$
Density functions : $f(x) ,f(t)$

1. Probability of an Event
$P(A)= {m \over n },$
Where
m is the number of possible positive outcomes ,
n is the total number of possible outcomes .
2. Range of Probability Values
$0 \le P(A) \le 1$
3. Certain Event
$P(A)=1$
4. Impossible Event
$P(A)=0$
5. Complement
$P(\bar {A})=1-P(A)$
6. Independent Events
$P(A/B)=P(A),$
$P(B/A)=P(B)$
7. Addition Rule for Independent Events
$P(A \cup B)=P(A)+P(B)$
8. Multiplication Rule for Independent Events
$P(A \cap B)= P(A).P(B)$
9. General Addition Rule
$P(A \cup B)=P(A)+P(B)-P(A \cap B) ,$
Where
$A \cup B$ is the union of events A and B ,
$A \cap B$ is the intersection of events A and B .
10. Conditional Probability
$P(A/B)= { P(A \cap B) \over P (B)}$
11. $P(A \cap B)=P(B).P(A/B)=P(A).P(B/A)$
12. Law of Total Probability
$P(A)= \displaystyle\sum_{i=1}^m P(B_i)P(A/B_i),$
Where $B_i$ is a sequence of mutually exclusive events.
13. Bayes' Theorem
$P(B/A)= {P(A/B).P(B) \over P(A)}$
14. Bayes' Formula
$P(B_i/A)= {P(B_i).P(A/B_i) \over \displaystyle\sum_{i=1}^m P(B_i).P(A/B_i)},$
Where
$B_i$ is a set of mutually exclusive events ( hypotheses ),
A is the final event,
$P(B_i)$ are the prior probabilities,
$P(B_i/A)$ are the posterior probabilities.
15. Law of Large Number
$P \Bigg (\Bigg|{S_n \over n}-\mu \Bigg | \ge \varepsilon \Bigg ) \rightarrow 0 \; \; as \; \; n \to \infty ,$
$P \Bigg (\Bigg|{S_n \over n}-\mu \Bigg | < \varepsilon \Bigg ) \rightarrow 1 \; \; as \; \; n \to \infty ,$
Where
$S_n$ is the sum of raandom variables ,
n is the number of possible outcomes .
16. Chebyshev Inequality
$P(|X- \mu| \ge \varepsilon ) \le {V(X) \over \varepsilon^2 }$,
Where $V(X)$ is the variance of X .
17. Normal Density Function
$\phi(x)= {1 \over \sigma \sqrt {2 \pi}}e^{-{(x-\mu)^2 \over 2 \sigma^2}}$
Where X is a particular Outcome  .
18. Standard Normal Density Function
$\phi(z)= {1 \over \sqrt {2 \pi}}e^{-{z^2 \over 2}}$
Average value $\mu = 0 ,$ deviation $\sigma =1.$ 19. Standard Z Value
$Z= {X- \mu \over \sigma}$
20. Cumulative Normal Distribution Function
$F(x)={1 \over \sigma \sqrt {2 \pi} } \int\limits_{- \infty}^x e^{-{(t-\mu)^2 \over 2\sigma^2} }dt ,$
Where
x is a paarticular outcome ,
t is a variable of integration.
21. $P( \alpha < X < \beta )= F \bigg ({\alpha - \mu \over \sigma } \bigg)-F \bigg({\beta - \mu \over \sigma } \bigg) ,$
Where
X is normally distributed random vaariable ,
F is cumulaative normal distribution function ,
$P( \alpha < X < \beta )$ is interval probability .
22. Cumulative Distribution Function
$F(x)=P(X<x) = \int\limits_{- \infty}^x f(t) dt,$
Where t is a variable of integration.
23. Bernoulli Trials Process
$\mu = np , \sigma^2=npq,$
Where
n is a sequence of experiments ,
p is the probability of success of each experiments,
q is the probability of failure , q=1-p.
24. Binomial Distribution Function
$b(n,p,q) =\bigg ( \substack {n \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ k} \bigg )p^kq^{n-k},$
$\mu =np \; , \sigma^2= {q \over p^2},$
Where
T is the first successful event is the series
j is the event numbeer ,
p is the probability that any one event is successful,
q is the probability of failure , q= 1-p.
25. Poisson Distribution
$P(X=k)\approx{\lambda^k \over k!} e^{- \lambda } , \lambda= np,$
$\mu = \lambda, \sigma^2= \lambda ,$
Where
$\lambda$ is the rate of occurrence ,
k is the number of positive outcomes .
26. Density Function
$P(a \le X \le b)= \int \limits_a^b f(x) dx$
27. Continuous uniform Density
$f={1 \over b-a}, \mu = {a+b \over 2 } ,$
where f is the density function .
28. Exponential Density Function
$f(t) = \lambda e^{-\lambda t} , \mu= \lambda , \sigma ^2= \lambda ^2$
where t is time , $\lambda$ is the failure rate .
29. Exponential Distribution Function
$F(t)=1-e^{-\lambda t},$
where t is time , $\lambda$ is the failure rate .
30. Expected value of discrete random Variables
$\mu = E(X) = \displaystyle\sum_{i=1}^n x_ip_i,$
where $x_i$ is a particular outcome , $p_i$ is its probability .
31. Expected Value of Continuous Random Variables
$\mu =E(X)= \int \limits_{- \infty}^\infty xf(x) dx$
32. $E(X^2)=V(X)+ \mu^2,$
where
$\mu=E(X)$ is the expected value ,
V(X) is the variance .
33. Markov Inequality
$P(X>k) \le {E(X) \over k},$
where k is some constant .
34. Variance of Discrete Random variables
$\sigma ^2 =V(X)=E \bigg [(X- \mu)^2 \bigg ]= \displaystyle \sum _{i=1} ^n ( x_i - \mu ) ^2 p_i ,$
where
$x_i$ is a particular outcome ,
$p_i$ is its probability .
35. Variance of Continuous Random Variables
$\sigma ^2 =V(X)=E \bigg [(X- \mu)^2 \bigg ]= \displaystyle \int\limits _{- \infty} ^\infty ( x - \mu ) ^2 f(x)dx$
36. Properties of Variance
$V(X+Y)=V(X)+V(Y),$
$V(X-Y)=V(X)+V(Y),$
$V(X+c)= V(X),$
$V(cX)=c^2V(X),$
where c is a constant .
37. Standard Deviation
$D(X)= \sqrt {V(X)}= \sqrt {E [(X- \mu )^2]}$
38. Covariance
$cov(X,Y)=E[(X- \mu(X)) (Y- \mu(Y))]=E(XY)-\mu(X)\mu(Y),$
Where
X is random variable ,
V(X) is the variance of X ,
$\mu$ is the expected value of X or Y .
39. Correlation
$p(X,Y)= { cov(X,Y) \over \sqrt {V(X)V(Y)}},$
Where
V(X) is the variance of X ,
V(Y) is the variance of Y.