Expected value

==Expected values of common distributions==
The following table gives the expected values of some commonly occurring [[probability distribution]]s. The third column gives the expected values both in the form immediately given by the definition, as well as in the simplified form obtained by computation therefrom. The details of these computations, which are not always straightforward, can be found in the indicated references.
{| class="wikitable"
!Distribution
!Notation
!Mean E(X)
|-
|[[Bernoulli distribution|Bernoulli]]{{sfnm|1a1=Casella|1a2=Berger|1y=2001|1p=89|2a1=Ross|2y=2019|2loc=Example 2.16}}
|<math>X \sim~ b(1,p)</math>
|<math>0\cdot(1-p)+1\cdot p=p</math>
|-
|[[Binomial distribution|Binomial]]{{sfnm|1a1=Casella|1a2=Berger|1y=2001|1loc=Example 2.2.3|2a1=Ross|2y=2019|2loc=Example 2.17}}
|<math>X \sim B(n,p)</math>
|<math>\sum_{i=0}^n i{n\choose i}p^i(1-p)^{n-i}=np</math>
|-
|[[Poisson distribution|Poisson]]{{sfnm|1a1=Billingsley|1y=1995|1loc=Example 21.4|2a1=Casella|2a2=Berger|2y=2001|2p=92|3a1=Ross|3y=2019|3loc=Example 2.19}}
|<math>X \sim \mathrm{Po}(\lambda)</math>
|<math>\sum_{i=0}^\infty \frac{ie^{-\lambda}\lambda^i}{i!}=\lambda</math>
|-
|[[Geometric distribution|Geometric]]{{sfnm|1a1=Casella|1a2=Berger|1y=2001|1p=97|2a1=Ross|2y=2019|2loc=Example 2.18}}
|<math>X \sim \mathrm{Geometric}(p)</math>
|<math>\sum_{i=1}^\infty ip(1-p)^{i-1}=\frac{1}{p}</math>
|-
|[[Uniform distribution (continuous)|Uniform]]{{sfnm|1a1=Casella|1a2=Berger|1y=2001|1p=99|2a1=Ross|2y=2019|2loc=Example 2.20}}
|<math>X\sim U(a,b)</math>
|<math>\int_a^b \frac{x}{b-a}\,dx=\frac{a+b}{2}</math>
|-
|[[Exponential distribution|Exponential]]{{sfnm|1a1=Billingsley|1y=1995|1loc=Example 21.3|2a1=Casella|2a2=Berger|2y=2001|2loc=Example 2.2.2|3a1=Ross|3y=2019|3loc=Example 2.21}}
|<math>X\sim \exp(\lambda)</math>
|<math>\int_0^\infty \lambda xe^{-\lambda x}\,dx=\frac{1}{\lambda}</math>
|-
|[[Normal distribution|Normal]]{{sfnm|1a1=Casella|1a2=Berger|1y=2001|1p=103|2a1=Ross|2y=2019|2loc=Example 2.22}}
|<math>X\sim N(\mu,\sigma^2)</math>
|<math>\frac{1}{\sqrt{2\pi\sigma^2}}\int_{-\infty}^\infty xe^{-(x-\mu)^2/2\sigma^2}\,dx=\mu</math>
|-
|[[Standard normal|Standard Normal]]{{sfnm|1a1=Billingsley|1y=1995|1loc=Example 21.1|2a1=Casella|2a2=Berger|2y=2001|2p=103}}
|<math>X\sim N(0,1)</math>
|<math>\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty xe^{-x^2/2}\,dx=0</math>
|-
|[[Pareto distribution|Pareto]]{{sfnm|1a1=Johnson|1a2=Kotz|1a3=Balakrishnan|1y=1994|1loc=Chapter 20}}
|<math>X\sim \mathrm{Par}(\alpha, k)</math>
|<math>\int_k^\infty\alpha k^\alpha x^{-\alpha}\,dx=\begin{cases}\frac{\alpha k}{\alpha-1}&\alpha>1\\ \infty&0 \leq \alpha \leq 1.\end{cases}</math>
|-
|[[Cauchy distribution|Cauchy]]{{sfnm|1a1=Feller|1y=1971|1loc=Section II.4}}
|<math>X\sim \mathrm{Cauchy}(x_0,\gamma)</math>
|<math>\frac{1}{\pi}\int_{-\infty}^\infty \frac{\gamma x}{(x - x_0)^2 + \gamma^2}\,dx</math> is [[indeterminate form|undefined]]
|}