Moment-generating function

The moment generating function of the - th moment of the probability distribution of a random variable can be determined. In probability theory, the moment generating function of a random variable is defined by

If their expected values ​​exist. Where and are the moments of a probability distribution.

• 3.1 Normal distribution
• 3.2 exponential

For continuous probability distribution

If a continuous probability density, then the moment generating function is by

Given, the - th moment of is. The term is therefore just the two-sided Laplace transform of the probability measure defined by.

Comments

Origin of the term of the moment generating function

The term torque producing refers to the fact that the -th derivative of the point 0 (zero) is equal to the -th moment of the random variable:

By specifying all non-vanishing moments of any probability distribution is completely determined if the moment generating function on an open interval exists.

Connection with the characteristic feature

The moment generating function is closely related to the characteristic function. It applies if the right-hand side of the equation exists. In contrast to the moment generating function of the characteristic function for arbitrary random variables exist.

Other generating functions

Other generating functions counting the characteristic function, the probability generating function and the kumulantenerzeugende function as the logarithm of the moment generating function.

Sums of independent random variables

The moment generating function of a sum of independent random variables is the product of their moment generating functions: are independent, then for

Uniqueness property

If the moment generating function of a random variable in an environment of finite, then it determines the distribution of unique. Formally, this means:

Let and be two random variables with moment generating functions and such that there is one with all. Then if and only if for all.

Examples

Normal distribution

The moment generating function of a normally distributed random variable is.

Exponential distribution

The moment generating function of an exponentially distributed random variable with parameter

Generalization to multi-dimensional random variables

The moment generating function can be expanded - dimensional real random vectors as follows:

Where the standard scalar called.