Probability - Probability Formulas 1

Jumat, 14 Agustus 2020

Probability Formulas 1

 

•        Normal Density Function

,

where x is a particular outcome.

 

•       Standard Normal Density Function

 Average value μ = 0, deviation σ = 1.

Figure

•        Standard Z Value

 

•        Cumulative Normal Distribution Function

,

where

x is a particular outcome,

t is a variable of integration.

 

•,

where

X is normally distributed random variable

F is cumulative normal distribution function,

P(α <  X < β) is interval probability.

 

•,

where

X is normally distributed random variable

F is cumulative normal distribution function.

 

•        Cumulative Distribution Function

   ,

where t is a variable of integration.

 

•        Bernoulli Trials Process

μ = np , σ² = 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.

 

•        Binomial Distribution Function

,

μ = np , σ² = npq,

f(x) = (q + pe^x)ⁿ,

where

n is the number of trials of selections,

p is the probability of success of each experiments,

q is the probability of failure, q = 1 – p

 

•        Geometric Distribution

P(T = j) = q^{j – 1}p,

,

where

T is the first successful event is the series,

j is the event number,

p is the probability that any one event is successful,

q is the probability of failure, q = 1 – p

 

•        Poisson Distribution

,

μ = λ, σ² = λ,

where

λ is the rate of occurrence,

k is the number of positive outcomes.

 

•        Density Function

•        Continuous Uniform Density

,

where f is density function.

 

•        Exponential Density Function

f(t) = λe ^{– λt} , μ = λ, σ² = λ²

where t is time, λ is the failure rate.

 

•        Exponential Distribution Function

Ft) = 1 – e ^{– λt} ,

where t is time, λ is the failure rate.

 

•        Expected Value of Discrete Random Variables

,

where x_i is a particular outcome, p_i is its probability.

 

•        Expected Value of Continuous Random Variables

 

•        Properties of Expectations

E(X + Y) = E(X) + E(Y),

E(X – Y) = E(X) – E(Y),

E(cX) = cE(X),

E(XY) = E(X) · E(Y)

where c is a constant.

 

•        E(X²) = V(X) + μ²,

where

μ = E(X) is the expected value,

V(X) is the variance.

 

•        Markov Inequality

,

where k is some constant.

 

•        Variance of Discrete Random Variables

,

where

x_i is a particular outcome,

p_i is its probability.

 

•        Variance of Continuous

 

•        Properties of Variables

V(X + Y) = V(X) + V(Y),

V(X – Y) = V(X) – V(Y),

V(X + c) = V(X),

V(cX) = c²V(X)

where c is a constant.

 

•        Standard Deviation

•        Covariance

cov(X, Y) = E[(X – μ(X)) (Y – μ(Y))] = E(XY) – μ(X)μ(Y),

where

X is random variable,

V is the variance of X,

Μ is the expected value of X or Y.

 

•        Correlation

,

where

V(X) is the variance of X,

V(Y) is the variance of Y.

 

 

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