Artificial Intelligence 🤖
Probability & Statistics
Common Probability Distributions

Probability Distributions

ModelPMF // PDF fX(x∣f_{X}(x \mid parameters ))RangeE[X]E[X]Var⁡[X]\operatorname{Var}[X]
UniformfX(x∣n)=1nf_{X}(x \mid n)=\frac{1}{n}x=1,…,nx=1, \ldots, nn+12\frac{n+1}{2}n2−112\frac{n^{2}-1}{12}
BernoullifX(x∣p)=p1(x=1)(1−p)1(x=0)f_{X}(x \mid p)=p^{1(x=1)}(1-p)^{1(x=0)}x=0,1x=0,1ppp(1−p)p(1-p)
BinomialfX(x∣n,p)=(nx)px(1−p)n−xf_{X}(x \mid n, p)=\left(\begin{array}{l}n \\ x\end{array}\right) p^{x}(1-p)^{n-x}x=0,1,…,nx=0,1, \ldots, nnpn pnp(1−p)n p(1-p)
PoissonfX(x∣n,p)=e−np(np)x/x!f_{X}(x \mid n, p)=e^{-n p}(n p)^{x} / x !x=0,1,…,nx=0,1, \ldots, nnpn pnpn p
(Binomial approximation)
PoissonfX(x∣λ)=e−λλx/x!f_{X}(x \mid \lambda)=e^{-\lambda} \lambda^{x} / x !x=0,1,2,…x=0,1,2, \ldotsλ\lambdaλ\lambda
(Average occurrences)
PoissonfX(x∣λ^,T)=e−λ^T(λ^T)xx!f_{X}(x \mid \hat{\lambda}, T)=\frac{e^{-\hat{\lambda} T(\hat{\lambda} T)^{x}}}{x !}x=0,1,2,…x=0,1,2, \ldotsλ^T\hat{\lambda} Tλ^T\hat{\lambda} T
(Average rate)fX(x∣μ,σ)=12πσ2e−(x−μ)2/2σ2f_{X}(x \mid \mu, \sigma)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-(x-\mu)^{2} / 2 \sigma^{2}}−∞<x<∞-\infty<x<\inftyμ\muσ2\sigma^{2}
ExponentialfZ(z)=12πe−z2/2f_{Z}(z)=\frac{1}{\sqrt{2 \pi}} e^{-z^{2} / 2}−∞<x<∞-\infty<x<\infty01
NormalfX(xd∣λ^)=λ^e−λ^xf_{X}(x d\mid \hat{\lambda})=\hat{\lambda} e^{-\hat{\lambda} x}x>0x>01λ^\frac{1}{\hat{\lambda}}1λ^2\frac{1}{\hat{\lambda}^{2}}

Summary of Probability Distributions and Properties. Up until this point in the course, we have considered rather general probability distributions for discrete and continuous random variables (i.e. PMF/PDF fX(x)f_{X}(x) ). Indeed, to be a probability distribution a function only needs to satisfy the general requirements that it is (1) nonnegative for all outcomes and (2) must sum/integrate to a total value of one. We have already seen several such functions exist that satisfy these rather simple criteria. For instance, the following quadratic function is a valid PDF:

fX(x)=19x2 for 0<x<3, and 0 otherwise f_{X}(x)=\frac{1}{9} x^{2} \quad \text { for } 0<x<3, \text { and } 0 \text { otherwise }

However, we find in practice that there is a much smaller space of functions that capture the probabilistic behaviour of phenomena in the 'real' world. This chapter is devoted to reviewing these common probability functions, which turn out to have very interesting results regarding the calculations of their expectations and variances in terms of model parameters.

Multivariate DistributionsCommon Continuous Distributions