# Index [[Binomial distribution]] [[Continuous probability distributions]] [[Discrete probability distributions]] [[Error function]] [[Exponential distributions]] [[Gaussian distribution]] [[Lorentzian distribution]] [[Probability density functions]] [Quasiprobability distributions](Quasiprobability%20distributions.md) --- # Basic concepts ## Probability distributions We find that [probabilities](Statistics%20and%20Probability%20(index).md#Probability) can be modeled as [functions](Analysis%20(index).md#Functions) of [events](Statistics%20and%20Probability%20(index).md#Events) called _probability distributions_ or _probability measures._^c3a77a A [function,](Analysis%20(index).md#Functions) $P(A)$ is a [probability distribution](Probability%20distributions%20(index).md#Probability%20distributions) if it satisfies the following properties that we accept as axioms: 1. $P(A)\geq 0$ for all [events,](Statistics%20and%20Probability%20(index).md#Events) $A.$ 2. $P(\Omega)=1$ where $\Omega$ denotes a [sample space.](Statistics%20and%20Probability%20(index).md#Sample%20spaces) ^a36c41 3. If the elements of the [set](Sets.md) of [events](Statistics%20and%20Probability%20(index).md#Events) $\{A_1,A_2,A_3...\}$ are [disjoint.](disjoint) then for a [partition,](Statistics%20and%20Probability%20(index).md#Partitions%20of%20sample%20spaces) $\bigcup_{i=1}^{\infty} A_i$$P\bigg(\bigcup_{i=1}^{\infty} A_i\bigg)=\sum_{i=1}^\infty P(A_i)$ ### Uniform probability distributions A _uniform probaility_ distribution is one for which [the sample space $\Omega$](probability%20distributions%20(index)#^a36c41) has a finite number of elements each with finite [probabilities.](Statistics%20and%20Probability%20(index).md#Probability) # Bibliography [Wasserman, L., _All of Statistics - A Concise Course in Statistical Inference_, Springer, 2003.](Wasserman,%20L.,%20All%20of%20Statistics%20-%20A%20Concise%20Course%20in%20Statistical%20Inference,%20Springer,%202003..md) #MathematicalFoundations/StatisticsAndProbability/ProbabilityDistributions #MathematicalFoundations/StatisticsAndProbability/ProbabilityDistributions/DiscreteProbabilityDistributions #MathematicalFoundations/StatisticsAndProbability/ProbabilityDistributions/ContinuousProbabilityDistributions #index #Bibliography