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- Thread starter arunma
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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.

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

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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!

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arunma said:

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

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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.

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.

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.

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.

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.

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