Deriving the MGF of a Normal Distribution

Oct 13, 2025 by windailys.com 120 views

Introduction

The moment generating function (MGF) is a powerful tool used in probability theory to study the properties of a random variable. In this article, we will derive the moment generating function of a normal distribution, which is a fundamental probability distribution in statistics. We will also use the MGF to obtain the mean and variance of the normal distribution.

What is the Moment Generating Function?

The moment generating function of a random variable $X$ is defined as:

M_X(t) = E[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} f(x) dx

where $f(x)$ is the probability density function (PDF) of $X$ .

Deriving the MGF of a Normal Distribution

Let $X \sim N(\mu, \sigma^2)$ . The PDF of a normal distribution is given by:

f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp \left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

Substituting this into the definition of the MGF, we get:

M_X(t) = E[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} \frac{1}{\sigma \sqrt{2\pi}} \exp \left(-\frac{(x-\mu)^2}{2\sigma^2}\right) dx

To evaluate this integral, we can use the following substitution:

u = \frac{x-\mu}{\sigma}

This gives:

du = \frac{1}{\sigma} dx

and:

x = \mu + \sigma u

Substituting these into the integral, we get:

M_X(t) = \int_{-\infty}^{\infty} e^{t(\mu + \sigma u)} \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2}\right) du

Simplifying this expression, we get:

M_X(t) = e^{t\mu} \int_{-\infty}^{\infty} e^{t\sigma u} \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2}\right) du

This is a standard Gaussian integral, which can be evaluated as:

\int_{-\infty}^{\infty} e^{t\sigma u} \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2}\right) du = e^{\frac{t^2\sigma^2}{2}}

Substituting this back into the expression for $M_X(t)$ , we get:

M_X(t) = e^{t\mu} e^{\frac{t^2\sigma^2}{2}}

This is the moment generating function of a normal distribution.

Obtaining the Mean and Variance

The moment generating function can be used to obtain the mean and variance of a random variable. The first and second derivatives of the MGF are given by:

M_X'(t) = \mu + \sigma^2 t

M_X''(t) = \sigma^2

Evaluating these at $t=0$ , we get:

M_X'(0) = \mu

M_X''(0) = \sigma^2

These are the mean and variance of the normal distribution.

Conclusion

In this article, we have derived the moment generating function of a normal distribution and used it to obtain the mean and variance of the distribution. The MGF is a powerful tool for studying the properties of a random variable, and it has many applications in statistics and probability theory.

References

Johnson, N. L., & Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions. Wiley.
Papoulis, A. (1984). Probability, random variables, and stochastic processes. McGraw-Hill.

Mathematical Derivations

Derivation of the MGF

Let $X \sim N(\mu, \sigma^2)$ . The PDF of a normal distribution is given by:

f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp \left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

Substituting this into the definition of the MGF, we get:

M_X(t) = E[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} \frac{1}{\sigma \sqrt{2\pi}} \exp \left(-\frac{(x-\mu)^2}{2\sigma^2}\right) dx

To evaluate this integral, we can use the following substitution:

u = \frac{x-\mu}{\sigma}

This gives:

du = \frac{1}{\sigma} dx

and:

x = \mu + \sigma u

Substituting these into the integral, we get:

M_X(t) = \int_{-\infty}^{\infty} e^{t(\mu + \sigma u)} \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2}\right) du

Simplifying this expression, we get:

M_X(t) = e^{t\mu} \int_{-\infty}^{\infty} e^{t\sigma u} \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2}\right) du

This is a standard Gaussian integral, which can be evaluated as:

\int_{-\infty}^{\infty} e^{t\sigma u} \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2}\right) du = e^{\frac{t^2\sigma^2}{2}}

Substituting this back into the expression for $M_X(t)$ , we get:

M_X(t) = e^{t\mu} e^{\frac{t^2\sigma^2}{2}}

Derivation of the Mean and Variance

The moment generating function can be used to obtain the mean and variance of a random variable. The first and second derivatives of the MGF are given by:

M_X'(t) = \mu + \sigma^2 t

M_X''(t) = \sigma^2

Evaluating these at $t=0$ , we get:

M_X'(0) = \mu

M_X''(0) = \sigma^2

Q: What is the moment generating function of a normal distribution?

A: The moment generating function of a normal distribution is given by:

M_X(t) = e^{t\mu} e^{\frac{t^2\sigma^2}{2}}

Q: How is the moment generating function used to obtain the mean and variance of a normal distribution?

A: The moment generating function can be used to obtain the mean and variance of a normal distribution by taking the first and second derivatives of the MGF. The first derivative is given by:

M_X'(t) = \mu + \sigma^2 t

Evaluating this at $t=0$ , we get:

M_X'(0) = \mu

The second derivative is given by:

M_X''(t) = \sigma^2

Evaluating this at $t=0$ , we get:

M_X''(0) = \sigma^2

Q: What is the significance of the moment generating function in statistics and probability theory?

A: The moment generating function is a powerful tool for studying the properties of a random variable. It can be used to obtain the mean and variance of a distribution, as well as higher moments. The MGF is also used in the study of limit theorems, such as the central limit theorem.

Q: Can the moment generating function be used to determine if a distribution is normal?

A: Yes, the moment generating function can be used to determine if a distribution is normal. If the MGF of a distribution is given by:

M_X(t) = e^{t\mu} e^{\frac{t^2\sigma^2}{2}}

then the distribution is normal with mean $\mu$ and variance $\sigma^2$ .

Q: How is the moment generating function used in practice?

A: The moment generating function is used in a variety of applications, including:

Hypothesis testing: The MGF is used to test hypotheses about the mean and variance of a distribution.
Confidence intervals: The MGF is used to construct confidence intervals for the mean and variance of a distribution.
Regression analysis: The MGF is used in regression analysis to model the relationship between a dependent variable and one or more independent variables.
Time series analysis: The MGF is used in time series analysis to model the behavior of a time series.

Q: What are some common applications of the moment generating function?

A: Some common applications of the moment generating function include:

Insurance: The MGF is used to model the distribution of insurance claims.
Finance: The MGF is used to model the distribution of stock prices and returns.
Engineering: The MGF is used to model the distribution of failure times for mechanical systems.
Biology: The MGF is used to model the distribution of population sizes and growth rates.

Q: What are some common mistakes to avoid when using the moment generating function?

A: Some common mistakes to avoid when using the moment generating function include:

Incorrectly assuming a normal distribution: The MGF is only valid for normal distributions.
Failing to check the assumptions: The MGF assumes that the distribution is continuous and has a finite variance.
Using the MGF without considering the underlying assumptions: The MGF is only valid if the underlying assumptions are met.

Q: What are some common extensions of the moment generating function?

A: Some common extensions of the moment generating function include:

Multivariate moment generating function: The MGF can be extended to multivariate distributions.
Time-varying moment generating function: The MGF can be extended to time-varying distributions.
Non-parametric moment generating function: The MGF can be extended to non-parametric distributions.

Q: What are some common software packages that implement the moment generating function?

A: Some common software packages that implement the moment generating function include:

R: The moments package in R implements the MGF for a variety of distributions.
Python: The scipy package in Python implements the MGF for a variety of distributions.
MATLAB: The stats package in MATLAB implements the MGF for a variety of distributions.