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Moment Generating Functions

Moment Generating Function (MGF) is a mathematical tool used in probability and statistics to summarize all the moments (like mean and variance) of a random variable. Think of it as a compact way to store and access important characteristics of a distribution.

Imagine you have a random variable representing the number of heads in a series of coin flips. The MGF of this variable can help you quickly find the mean number of heads and other properties without doing all the tedious calculations manually.

In this article, we will discuss the concept of MGF(Moment Generating Functions) in detail including examples of MGF of various distributions.

Table of Content

What is a Moment Generating Function?

Moment Generating Function (MGF) is a function used in probability theory and statistics to express the moments of a random variable. Moments are quantitative measures related to the shape of the random variable’s probability distribution, such as

Moment generating function (MGF) is used to summarize the distribution of a random variable. It is particularly useful for proving the convergence in distribution.

Definition of Moment Generating Function

The moment generating function MX(t) of a random variable X is defined as:

MX(t) = E[etX]

Where:

  • MX(t) is the moment generating function of X.
  • E denotes the expected value.
  • t is a real number.

for all values of t in some neighborhood of 0 where the expectation exists.

Moment Generating Functions of Common Distributions

Moment Generating Function for some of the most common distributions are listed in the following table:

Distribution

Moment Generating Function

Bernoulli Distribution

[Tex]M_X(t) = p e^t + (1 – p)[/Tex]

Binomial Distribution

[Tex]M_X(t) = (p e^t + (1 – p))^n[/Tex]

Geometric Distribution

[Tex]M_X(t) = \frac{p e^t}{1 – (1 – p) e^t}, \quad t < -\ln(1 – p)[/Tex]

Poisson Distribution

[Tex]M_X(t) = \exp(\lambda (e^t – 1))[/Tex]

Uniform Distribution

[Tex]M_X(t) = \frac{e^{tb} – e^{ta}}{t(b – a)}, \quad t \neq 0[/Tex]

Exponential Distribution

[Tex]M_X(t) = \frac{\lambda}{\lambda – t}, \quad t < \lambda[/Tex]

Normal Distribution

[Tex]M_X(t) = \exp(\mu t + \frac{\sigma^2 t^2}{2})[/Tex]

Gamma Distribution

[Tex]M_X(t) = (1 – \frac{t}{\theta})^{-k}, \quad t < \theta[/Tex]

Beta Distribution

[Tex]M_X(t) = \sum_{n=0}^\infty \frac{t^n}{n!} \frac{\Gamma(\alpha + n) \Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\alpha + \beta + n)}, \quad |t| < 1[/Tex]

Chi-Square Distribution

[Tex]M_X(t) = (1 – 2t)^{-k/2}, \quad t < \frac{1}{2}[/Tex]

Student’s t Distribution

Not defined

Weibull Distribution

No simple closed form

Moments from Moment Generating Functions

Moments of a random variable can be derived from its Moment Generating Function (MGF). The nth moment of a random variable X is given by taking the nth derivative of the MGF with respect to t and evaluating it at t = 0:

[Tex]\mathbb{E}[X^n] = M_X^{(n)}(0)[/Tex]

DistributionMean (E[X])Variance (Var(X))
Bernoulli Distributionpp(1−p)
Binomial Distributionnpnp(1−p)
Geometric Distribution1/p(1 − p)/p2
Poisson Distributionλλ
Uniform Distribution(a + b)/2​(b − a)2/12​
Exponential Distribution1/λ1/λ2
Normal Distributionμσ2
Gamma Distributionkθkθ2
Chi-Square Distributionk2k

Properties of Moment Generating Functions

Some of the common properties of moment generating functions are:

Existence

The MGF MX(t) exists for all values of t in an open interval containing 0. If the MGF exists for some t in this interval, it will be finite for all t in this interval.

Uniqueness

If two random variables have the same MGF in an open interval around t = 0, they have the same probability distribution. This property makes MGFs a powerful tool for characterizing distributions.

Additivity for Independent Variables

For two independent random variables X and Y, the MGF of their sum Z = X + Y is the product of their individual MGFs: MZ(t) = MX(t) MY(t). This property extends to any finite number of independent random variables.

MGF of a Linear Combination

If X is a random variable and a and b are constants, the MGF of the linear transformation Y = aX + b is given by:

[Tex]M_Y(t) = e^{bt} M_X(at)[/Tex]

Moment, Cumulant and Probability Generating Function

Moment, Cumulant, and Probability Generating Functions are three different mathematical tools used in probability theory and statistics to characterize and analyze the properties of random variables and their distributions.

Each function provides unique insights into the underlying distribution, helping to simplify the computation and understanding of moments, cumulants, and probabilities.

Cumulant Generating Function (CGF)

Cumulant Generating Function (CGF) is another important function in probability theory, which is closely related to the Moment Generating Function (MGF). The CGF, denoted as KX(t), is the natural logarithm of the MGF:

[Tex]K_X(t) = \ln(M_X(t))[/Tex]

The nth cumulant of a random variable X is given by the nth derivative of the CGF evaluated at t = 0:

[Tex]\kappa_n = \frac{d^n K_X(t)}{dt^n} \Bigg|_{t=0}[/Tex]

Cumulants are used to describe the shape of the probability distribution, similar to moments, but they have properties that often make them more convenient for certain types of statistical analysis, especially when dealing with sums of random variables.

Probability Generating Function (PGF)

Probability Generating Function (PGF) is another useful tool in probability theory, particularly for discrete random variables. The PGF of a discrete random variable X is defined as:

[Tex]G_X(s) = \mathbb{E}[s^X] = \sum_{k=0}^{\infty} P(X = k) s^k[/Tex]

MGF, CGF, and PGF of common distributions are listed in the following table:

DistributionMGFCGFPGF
Bernoulli

[Tex]M_X(t) = p e^t + (1 – p)[/Tex]

[Tex]K_X(t) = \ln(p e^t + (1 – p))[/Tex]

[Tex]G_X(s) = p s + (1 – p)[/Tex]

Binomial

[Tex]M_X(t) = (p e^t + (1 – p))^n[/Tex]

[Tex]K_X(t) = n \ln(p e^t + (1 – p))[/Tex]

[Tex]G_X(s) = (p s + (1 – p))^n[/Tex]

Geometric

[Tex]M_X(t) = \frac{p e^t}{1 – (1 – p) e^t}, \quad t < -\ln(1 – p)[/Tex]

[Tex]K_X(t) = \ln\left(\frac{p e^t}{1 – (1 – p) e^t}\right)[/Tex]

[Tex]G_X(s) = \frac{p s}{1 – (1 – p) s}[/Tex]

Poisson

[Tex]M_X(t) = \exp(\lambda (e^t – 1))[/Tex]

[Tex]K_X(t) = \lambda (e^t – 1)[/Tex]

[Tex] G_X(s) = \exp(\lambda (s – 1))[/Tex]

Uniform

[Tex]M_X(t) = \frac{e^{tb} – e^{ta}}{t(b – a)}, \quad t \neq 0[/Tex]

[Tex]K_X(t) = \ln\left(\frac{e^{tb} – e^{ta}}{t(b – a)}\right)[/Tex]

N/A
Exponential

[Tex]M_X(t) = \frac{\lambda}{\lambda – t}, \quad t < \lambda[/Tex]

[Tex]K_X(t) = -\ln\left(1 – \frac{t}{\lambda}\right)[/Tex]

[Tex]G_X(s) = \frac{\lambda s}{\lambda – (s – 1)}[/Tex]

​

Conclusion

Moment Generating Functions (MGFs) are an essential tool in probability and statistics, providing a compact and efficient way to derive the moments of a random variable and to characterize its distribution. By using the MGF, one can easily find the mean, variance, and higher-order moments of a distribution, simplifying many statistical analyses.

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FAQs on Moment Generating Functions

Define moment generating function.

A Moment Generating Function (MGF) is a function that summarizes all moments of a random variable, defined as [Tex]M_X(t) = E[e^{tX}][/Tex], where E denotes the expected value.

What is MGF?

MGF is the short form for moment generating function.

What is MGF of binomial distribution?

The MGF of a binomial distribution with parameters n and p is [Tex]M_X(t) = [(1 – p) + p e^t]^n[/Tex].

What is MGF for normal distribution?

The MGF of a normal distribution with mean (μ) and variance (σ2) is [Tex]M_X(t) = \exp(\mu t + \frac{1}{2}\sigma^2 t^2)[/Tex].

Write MGF for Bernoulli Distribution.

The MGF of a Bernoulli distribution with parameter p is [Tex]M_X(t) = (1 – p) + p e^t[/Tex].


 


Reffered: https://www.geeksforgeeks.org

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