If the moment generation function is the integral of e^tx.fx(x).dx

In summary, the moment generation function is a mathematical concept used to describe the relationship between a probability distribution and its moments. It allows us to calculate the moments of a distribution and is related to the probability distribution through derivatives. In some cases, it can be used to find the probability distribution, but it does have limitations, such as not existing for all distributions and being difficult to calculate for complex distributions.
  • #1
RufusDawes
156
0
from zero to infinity,

and the first raw moment about zero is the first derivative of the mgf evaluated at t=0... then why do we need to integrate the function ?

Wouldn't the first raw moment just be e^tx.fx(x) dx, i.e the derivative of the function we just integrated ?

Why do we integrate and then find the derivative of the same function ?
 
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  • #2
No. You aren't paying attention to which variable is used in the integration and which variable is used in the differentiation. Look at some specific examples.
 

Related to If the moment generation function is the integral of e^tx.fx(x).dx

1. What is the moment generation function?

The moment generation function is a mathematical concept used in probability and statistics to describe the relationship between a probability distribution and its moments. It is defined as the integral of the exponential function etx multiplied by the probability density function f(x) and integrated over all possible values of x.

2. What is the purpose of the moment generation function?

The moment generation function allows us to calculate the moments of a probability distribution, such as the mean, variance, and higher order moments, by taking derivatives with respect to t. This provides a convenient way to characterize and compare different distributions.

3. How is the moment generation function related to the probability distribution?

The moment generation function is a function of the parameter t and is unique to each probability distribution. By taking the appropriate derivatives, we can recover the moments of the distribution. Conversely, if we know the moments of a distribution, we can find the moment generation function.

4. Can the moment generation function be used to find the probability distribution?

Yes, in some cases, the moment generation function can be inverted to find the probability distribution. This is known as the inverse moment problem. However, this is not always possible and may require additional information about the distribution.

5. Are there any limitations to using the moment generation function?

While the moment generation function is a useful tool in probability and statistics, it does have its limitations. It may not exist or be defined for all distributions, and even when it does, it may not provide an intuitive understanding of the distribution. Additionally, the moment generation function may be difficult to calculate for complex distributions.

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