# Moment Generating Functions and Probability Density Functions

• arunma
In summary, the conversation discusses the relationship between moment generating functions and probability density functions. It is mentioned that the Fourier transform of the density function can be obtained from the moments, and the inverse transform can retrieve the density function. However, for distribution functions without a density, the process is more complicated. The conversation also mentions a specific Fourier transform and its use in retrieving a probability density function.

#### arunma

I was reading that moment generating functions have the property of uniqueness. So just wondering: is there a way to get a probability density function from a moment generating function?

The Fourier transform of the density function (called the characterictic function) can be obtained from the moments. The inverse transform of the ch. func. will give you the density function back. For distribution functions without a density, it is a little more complicated.

note: If the mgf exists in a neighborhood around 0 then the characteristic function = mgf(i*t)

Fourier Transforms of sinh

Hello:

I am referring to 'Table of Laplace Transforms' by Roberts&Kaufman. But I cannot seem to get a soln for the following Fourier Transform to retrieve my probability density f(x)

c2 * Integral{e^(iwx) * sinh[sqrt(2w)c1] / sinh[sqrt(2w)pi] dw} = f(x)

where -pi< c1 <=0 and c2 is a constant that scales the integral appropriately so that f(x) is p.d.f. Thanks for your help!

arunma said:
I was reading that moment generating functions have the property of uniqueness. So just wondering: is there a way to get a probability density function from a moment generating function?

In general, moment generating functions DO NOT have the property of uniqueness. C.F. s are unique.

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## 1. What is a moment generating function (MGF)?

A moment generating function (MGF) is a mathematical tool used to describe the probability distribution of a random variable. It is a function that generates moments of a random variable, which can provide useful information such as mean, variance, and higher moments.

## 2. How is a moment generating function related to a probability density function (PDF)?

The MGF and PDF are closely related as the MGF is the Laplace transform of the PDF. This means that the MGF contains all the information about the distribution of the random variable in its moments, while the PDF describes the shape of the distribution.

## 3. What is the purpose of using a moment generating function in statistics?

The moment generating function is a useful tool in statistics as it allows for the calculation of moments of a random variable without having to use complex integrals. It also simplifies the process of finding the distribution of a sum of independent random variables, as the MGF of the sum is the product of the individual MGFs.

## 4. Can the moment generating function be used for all types of probability distributions?

No, the moment generating function can only be used for certain types of probability distributions, such as the normal, exponential, and gamma distributions. It cannot be used for discrete distributions, such as the binomial or Poisson distributions.

## 5. How can the moment generating function be used to find the probability of a specific event?

The MGF can be used to find the probability of a specific event by taking the derivative of the MGF to obtain the probability density function, and then using this function to calculate the desired probability. This method is particularly useful for finding the probability of rare events, as it involves less computation than other methods.