= Expected Value =

The '''expected value''' of a random variable is its average, i.e. the value that can be assumed for a finite set of observations without impacting the sum.

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== Description ==

For a random variable ''X'', the expected value is usually notated as ''E[X]'', ''E(X)'', ''X̅''. 

More commonly though, a particular distribution function is assumed and the expected value is written in terms appropriate for the corresponding [[Analysis/Moment#First_Moment|first moment]]. For example:
 * for a [[Analysis/NormalDistribution|Normal distribution]], ''μ''
 * for a [[Analysis/BernoulliDistribution|Bernoulli distribution]], ''p''
 * for a [[Analysis/BinomialDistribution|Binomial distribution]], ''np''
 * for a [[Analysis/PoissonDistribution|Poisson distribution]], ''λ''
 * for a [[Analysis/UniformDistribution|Uniform distribution]], ''(a+b)/2''
 * and so on...

The expected value of a random vector ''X'' is equal to the expected value of each member. This should make clear that the expected value is a vector of equal dimensions to the random vector itself.



=== Estimation ===

Definitionally, [[Statistics/SamplingDistribution|sample mean]] ''X̅'' is '''always''' an unbiased estimator for the expected value ''E[X]'', '''unless''' the distribution has no defined expected value at all (e.g. the [[Analysis/CauchyDistribution|Cauchy distribution]]).

The standard error of that estimator is a function of the population distribution's [[Analysis/Variance|variance]] and the sample size: ''σ^2^,,X̅,, = σ^2^,,X,,/n''.



=== Properties ===

Expectations are linear. If the expected value of a random variable ''X'' is known, adding a constant to ''X'' also adds that constant to the expected value. Multiplying ''X'' by a constant factor also multiplies the known variance by that factor. Succinctly: ''E[aX + b] = a E[X] + b''.

This holds for linear transformations of random vectors. Let ''x'' be a random vector of size ''n x 1'', ''b'' be a constant vector of the same size, and ''A'' be a constant matrix of size ''m x n''. Given these, ''E[Ax + b] = A E[x] + b''.

The expected value of the sum of two random variables (or vectors) ''X'' and ''Y'' is the sum of their expected values: ''E[X + Y] = E[X] + E[Y]''.

Similarly, if two random variables (or vectors) are independent, then the expected value of their product is the product of their expected values: ''E[XY] = E[X]E[Y]''.

Also consider the ratio of two independent random variables: ''E[X/Y] = E[X]E[1/Y]''. Note that this is ''not'' equivalent to ''E[X]/E[Y]''.

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== Formulation ==

Formally, the first moment of a continuous quantitative (numeric) random variable is:

{{attachment:cont1.svg}}

where ''Ω'' is the sample space and ''f'' is the probability density function for that distribution.

And for a discrete quantitative random variable:

{{attachment:dis1.svg}}

where ''Ω'' is the set of possible values and ''f'' is the probability mass function for that distribution (although the distributions of discrete random variables are commonly given as tables, rather than mathematical functions). 

When working with a distribution given as a table, pairing values of ''X'' with a known probability (i.e. there is a function ''p'' describing the probability of ''X'' taking on the value ''x,,i,,''), the most common formulation is:

{{attachment:exp1.svg}}

When working with a set of observations ''X'', where for each observation ''i'' there is an observed value ''x,,i,,'', the most common formula is:

{{attachment:exp2.svg}}

This can be equivalently expressed using a ones-vector ''j''. Let ''x'' be the vector of observed values:
 * ''(x⃗ · ĵ) 1/n''
 * ''(x^T^j) 1/n'' for a more [[LinearAlgebra|linear algebraic]] formulation
 * ''(x^T^Ij) 1/n'' to look more like a [[LinearAlgebra/QuadraticForm|quadratic form]]



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