# Moments of normal distribution

• A
• senobim
In summary, the conversation discusses the calculation of the characteristic function of a normal distribution and the use of Taylor series to find the moments. The moments can be evaluated by taking the n-th derivative of the Taylor series and evaluating it at zero. The first derivative of the term representing the first moment is evaluated at zero, resulting in the value of ia. The conversation also addresses the presence of the i term in the derivative.
senobim
I have calculated characteristic function of normal distribution $$f_{X}(k)=e^{(ika-\frac{\sigma ^{2}k^{2}}{2})}$$ and now I would like to find the moments, so I know that you could expand characteristic function by Taylor series

$$f_{X}(k)=exp(1+\frac{1}{1!}(ika - \frac{\sigma^2k^2}{2})+\frac{1}{2!}(ika - \frac{\sigma^2k^2}{2})^2+\frac{1}{3!}(ika - \frac{\sigma^2k^2}{2})^3+...)$$

$$f_{X}(k)=exp(1+\frac{(ik)}{1!}\left \langle X^1 \right \rangle+\frac{(ik)^2}{2!}\left \langle X^2 \right \rangle+\frac{(ik)^3}{3!}\left \langle X^3 \right \rangle+...)$$

and the moments will be
$$\left \langle X^n \right \rangle$$

Now the problem is that I completely forgot how to evaluate Taylor series.
Could you help me to calculate for example second moment? I know what the answer should be, but I couldn't get it right.

Last edited:
Take the n-th derivative of the taylor series and evaluate at zero.
BTW your exponential function should not appear within the taylor expansion... this expansion IS the exponential funcition.

senobim
I am trying to calculate first derivate of term <x1>

$$\frac{d}{dk}(ika-\frac{\sigma^2k^{2} }{2})=(ia - \sigma ^{2}k)$$

now I am evaluating it at 0

$$f_{X}(0)= ia$$

And what will happen with a i term?

$f'_X(0)=iE(X)$, not E(X).

senobim
Thank you!

## 1. What are moments of normal distribution?

The moments of normal distribution refer to statistical measures that describe the shape, spread, and central tendency of a normal distribution. These moments are calculated using the data's mean, variance, and higher-order moments.

## 2. How many moments are there in a normal distribution?

There are an infinite number of moments in a normal distribution, with each moment representing a different aspect of the distribution's shape and behavior. The first four moments (mean, variance, skewness, and kurtosis) are the most commonly used in statistical analysis.

## 3. What is the first moment of a normal distribution?

The first moment of a normal distribution is the mean, which represents the central tendency or average value of the data. It is calculated by summing all the data points and dividing by the total number of points in the distribution.

## 4. How are moments used in data analysis?

Moments are used in data analysis to describe and understand the characteristics of a normal distribution. They can help identify outliers, assess the symmetry and peakiness of the distribution, and compare different distributions.

## 5. Can moments be used to identify a normal distribution?

Yes, moments can be used to identify a normal distribution by providing information about its shape and characteristics. For example, a normal distribution will have a skewness of 0 and a kurtosis of 3, while non-normal distributions will have different values for these moments.

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