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.