What is a moment generating function?

The MGF of a random variable X is M(t) = E[e^(tX)]. When it exists, the n-th derivative of M(t) evaluated at t=0 equals the n-th raw moment E[X^n]. This makes computing mean, variance, and higher moments algebraically straightforward.

Why does the MGF not always exist?

The MGF requires E[e^(tX)] to be finite in an open interval around t=0. Heavy-tailed distributions like the Cauchy have no MGF because their exponential moments diverge. All eight distributions in this calculator have well-defined MGFs within their valid parameter ranges.

What is excess kurtosis?

Excess kurtosis = kurtosis - 3. A normal distribution has excess kurtosis of 0. Positive excess kurtosis (leptokurtic) indicates heavier tails than normal; negative (platykurtic) means lighter tails. This calculator reports excess kurtosis throughout.

What is the difference between skewness and kurtosis?

Skewness measures asymmetry: a right-skewed distribution has a longer right tail (positive skewness). Kurtosis measures tail heaviness relative to the normal distribution. Together they describe the shape of a distribution beyond mean and variance.

How accurate are the computed values?

Values are computed using exact closed-form formulas derived from each distribution's MGF — there is no numerical approximation involved. Floating-point rounding may affect the last one or two decimal places for extreme parameter values.

Which distributions are supported?

Normal, Binomial, Poisson, Exponential, Gamma, Bernoulli, Geometric, and Uniform distributions are currently supported. Each has a well-known MGF with closed-form moment expressions.

Moment Generating Function Calculator

Select a probability distribution, enter its parameters, and instantly get the mean, variance, skewness, and kurtosis derived from its moment generating function.

Distribution & Parameters

Distribution

MGF Formula

Select a distribution and click Compute Moments to see results.

Mean (1st moment)

E[X]

Variance (2nd central)

Var(X) = E[X²] − (E[X])²

Std Deviation

σ = √Var(X)

Skewness (3rd std)

μ₃ / σ³

Excess Kurtosis (4th std − 3)

0 = normal tails; positive = heavier tails

Moment	Value	Notation

Summary

The moment generating function (MGF) of a random variable X is defined as M(t) = E[e^(tX)]. By taking successive derivatives of M(t) at t=0, you obtain the raw moments of the distribution — the first moment is the mean, the second central moment is the variance, and so on. This calculator applies closed-form MGF derivatives for eight common distributions so you get exact moment values instantly without numerical integration.

How it works

Select a probability distribution from the dropdown menu.
Enter the required parameter values for that distribution.
Click "Compute Moments" to evaluate the MGF-derived formulas.
Read off the mean, variance, standard deviation, skewness, and excess kurtosis.
Use the MGF formula panel to see the symbolic expression used.

Use cases

Verify distribution parameters in statistics homework or exams.
Quickly compare spread and shape across different distributions.
Teaching moment generating functions in probability courses.
Sanity-check simulation outputs against theoretical moments.
Compute skewness and kurtosis for risk and actuarial models.
Explore how parameter changes affect distribution shape.

Frequently Asked Questions

Last updated: 2026-05-28 · Reviewed by Nham Vu