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]]
|}
Description	What you type	What you get
Italic	''Italic text''	Italic text
Bold	'''Bold text'''	Bold text
Bold & italic	'''''Bold & italic text'''''	*Bold & italic text*
Description	What you type	What you get
Reference	Page text.<ref>[https://www.example.org/ Link text], additional text.</ref>	Page text.^[1]
Named reference	Page text.<ref name="test">[https://www.example.org/ Link text]</ref>	Page text.^[2]
Additional use of the same reference	Page text.<ref name="test" />	Page text.^[2]
Display references	<references />	↑ Link text, additional text. ↑ Link text