# Raw moments of Gaussian Distribution

• SeriousNoob
In summary, the conversation discusses the moments of a Gaussian Distribution. The first four moments are provided, and the topic of finding the 8th moment is brought up. It is mentioned that there is a property of the Gaussian distribution where all central moments above 2 are 0, making it unnecessary to calculate them using integrals. The method for calculating the third moment is also explained, and it is shown that the 8th moment can be expressed as a combination of the mean and standard deviation.
SeriousNoob
I'm wondering if there was a table of moments for a Gaussian Distribution, I found one up to the fourth moment
$$U \sim N(\mu, \sigma^2)$$
$$E[U^2]=\mu^2+\sigma^2$$
$$E[U^3]=\mu^3+3\mu\sigma^2$$
$$E[U^4]=\mu^4+6\mu\sigma^2+3\sigma^4$$

I'm doing a problem right now and i need the 8th moment.

It is a straightforward (tedious) integral.

You do not need to do integrals if you know the property of the Gaussian distribution that all central moments above 2 are 0. But I'm not saying it is the easiest method. Here is how you do it for $$m_3$$:

$$m_3 = \left< (x-<x>)^2 \right> = <x^3> - 3<x>^3 + 3<x><x^2> + <x>^3 = <x^3> + <x>^3 + 3 <x> ( <x^2> - <x>^2) = m_3 + \mu^3 + 3<x>\sigma^2 = 0$$

From this we get:

$$m_3 = \mu \sigma^2 + \mu^3$$

And so on...

Last edited:

## 1. What is a raw moment of Gaussian Distribution?

A raw moment of Gaussian Distribution is a mathematical concept that measures the shape and spread of a normal distribution. It is calculated by taking the integral of x^n multiplied by the probability density function of a normal distribution, where n is the desired moment.

## 2. How are raw moments of Gaussian Distribution used in statistics?

Raw moments of Gaussian Distribution are used to calculate higher order moments, such as variance and skewness, which provide important information about the shape and spread of a dataset. They are also used in hypothesis testing and in the calculation of confidence intervals.

## 3. What is the difference between raw moments and central moments of Gaussian Distribution?

The main difference between raw moments and central moments of Gaussian Distribution is the way they are calculated. Raw moments are calculated using the original data points, while central moments are calculated using the mean of the data. This means that central moments are less affected by extreme values in the dataset.

## 4. Can raw moments of Gaussian Distribution be negative?

Yes, raw moments of Gaussian Distribution can be negative. This is because they are calculated by taking the integral of x^n multiplied by the probability density function, which can result in negative values depending on the value of n.

## 5. How can I calculate raw moments of Gaussian Distribution?

You can calculate raw moments of Gaussian Distribution by taking the integral of x^n multiplied by the probability density function of a normal distribution, where n is the desired moment. Alternatively, you can use statistical software or online calculators to calculate raw moments for a given dataset.

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