Moment generating function help?

In summary, a moment generating function (MGF) is a mathematical function used to describe the probability distribution of a random variable. It is defined as the expected value of e^tx, where t is a real number and x is the random variable. It differs from a probability generating function in that it is used for continuous distributions and can find higher order moments. The MGF is important for finding moments of a distribution and can be calculated using the PDF of the random variable. However, it can only be used for certain types of distributions and not for discrete distributions.
  • #1
millwallcrazy
14
0
I know that hte MGF is = the E[e^tx]

How do i show that if i take a sample (X1;X2; : : :Xn) from the exponential density f(x) = A*e^(-Ax), then the sum Z = sum(Xi) has the gamma density?

I found that the MGF for the exponential was A/(t-A) if that helps

Thanks
 
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  • #2
Note that

[tex]
e^{t\sum X_i}} = \prod{e^{tX_i}}
[/tex]

This will relate the form of the mgf of the sum to the individual mgfs of the sample.
 

1. What is a moment generating function?

A moment generating function (MGF) is a mathematical function that is used to describe the probability distribution of a random variable. It is defined as the expected value of e^tx, where t is a real number and x is the random variable.

2. How is a moment generating function different from a probability generating function?

A moment generating function is used to describe the distribution of a random variable, while a probability generating function is used to describe the distribution of a discrete random variable. Additionally, the moment generating function can be used to find the moments of a distribution, while the probability generating function cannot.

3. Why is the moment generating function important?

The moment generating function is important because it provides a way to find the moments of a probability distribution, such as the mean and variance. It also allows for the calculation of higher order moments, which can be useful in statistical analysis and hypothesis testing.

4. How do you find the moment generating function of a distribution?

To find the moment generating function of a distribution, you first need to know the probability distribution function (PDF) of the random variable. Then, you can use the formula M(t) = E[e^tx] = ∫e^tx * f(x)dx, where f(x) is the PDF of the random variable, to calculate the moment generating function.

5. Can the moment generating function be used for all types of distributions?

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

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