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ExpectedValue

ExpectedValue[f, list]
gives the expected value of the pure function f with respect to the values in list.

ExpectedValue[f, list, x]
gives the expected value of the function f of x with respect to the values of list.

ExpectedValue[f, dist]
gives the expected value of the pure function f with respect to the symbolic distribution dist.

ExpectedValue[f, dist, x]
gives the expected value of the function f of x with respect to the symbolic distribution dist.

MORE INFORMATION

For the list ${x_1,x_2,...,x_n}$ , the expected value of f is given by .

For a continuous distribution dist, the expected value of f is given by where is the probability density function of dist and the integral is taken over the domain of dist.

For a discrete distribution dist, the expected value of f is given by where is the probability mass function of dist and summation is over the domain of dist.

The following option can be given:

assumptions to make about parameters

Basic Examples (3)

Find the expected value of

in a Poisson distribution:

Use a pure function:

Expected value for a list:

Find the expected value of

in a Poisson distribution:

Use a pure function:

Expected value for a list:

Compute the expected value of any function:

Do the computation numerically:

Obtain expectations with conditions:

Obtain results correct for given assumptions on symbols:

Applications (2)

Obtain the raw moments of a distribution:

Construct a mixture density, here a Poisson-inverse Gaussian mixture:

Properties & Relations (7)

ExpectedValue of a function is the integral or sum of that function times the PDF:

ExpectedValue of

for real t is the CharacteristicFunction:

ExpectedValue of a constant is the constant:

ExpectedValue of a random variable is the Mean:

ExpectedValue of the squared difference from the Mean is the Variance:

ExpectedValue for a list is a Mean:

CentralMoment is equivalent to an expected value:

CentralMoment Mean Variance Skewness Kurtosis CharacteristicFunction Integrate

Discrete Distributions

Continuous Distributions

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