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Probability and Distributions Cheat Sheet

SymbolConstantValueUnit
Pr(a<X<b)Continuous interval probabilityPr(a<X<b) = integral(a..b) p(x) dxformula
P(a)CDFP(a) = integral(-infinity..a) p(x) dxformula
E[g(X)]Discrete expectationE[g(X)] = sum_x g(x)Pr(X=x)formula
E[g(X)]Continuous expectationE[g(X)] = integral(-infinity..infinity) g(x)p(x) dxformula
Var(X)VarianceVar(X) = E[X^2] - E[X]^2formula
sigmaStandard deviationsigma = sqrt(Var(X))formula
Pr(A union B)Union rulePr(A union B)=Pr(A)+Pr(B)-Pr(A intersection B)formula
Pr(A intersection B)Independent intersectionPr(A intersection B)=Pr(A)Pr(B) when A and B are independentformula
Pr(A|B)Conditional probabilityPr(A|B)=Pr(A intersection B)/Pr(B)formula
E[X+Y]Expectation of sumE[X+Y]=E[X]+E[Y]formula
E[cX]Scaling expectationE[cX]=cE[X]formula
Pr(A_i|B)Bayes theoremPr(A_i|B)=Pr(B|A_i)Pr(A_i)/sum_j Pr(A_j)Pr(B|A_j)formula
Pr(X=k)Binomial probabilityPr(X=k)=C(n,k)p^k(1-p)^(n-k)formula
E[X]Binomial meanE[X]=npformula
Pr(X=k)Poisson probabilityPr(X=k)=exp(-lambda)*lambda^k/k!formula
E[X]Poisson meanE[X]=lambdaformula
p(x)Normal densityp(x)=exp(-(x-mu)^2/(2sigma^2))/(sqrt(2pi)*sigma)formula
Pr(X=k)Geometric probabilityPr(X=k)=p(1-p)^(k-1)formula
E[X]Geometric meanE[X]=1/pformula
E[T]Coupon collector meanE[T]=nH_nformula
Pr(|X|>=lambda)Markov inequalityPr(|X|>=lambda)<=E[|X|]/lambdaformula
Pr(|X-E[X]|>=lambda*sigma)Chebyshev inequalityPr(|X-E[X]|>=lambda*sigma)<=1/lambda^2formula

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A searchable formula reference transcribed and normalized from the supplied Theoretical Computer Science Cheat Sheet.

Probability and Distributions formula reference

This searchable reference organizes formulas from the supplied Theoretical Computer Science Cheat Sheet into web-friendly notation. Use the search box to find a formula by name or symbol, then review its assumptions before applying it.

Some identities use conventional mathematical notation such as summation limits, factorials, congruences, expectations and asymptotic bounds. These entries are reference material and should be interpreted within their stated domains.