Unique properties of Gaussians

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In summary, Gaussians, also known as Gaussian distributions or normal distributions, are unique because they have a bell-shaped curve and are symmetric around the mean. Their main properties include a well-defined mean and standard deviation, a finite range, and a total area under the curve equal to 1. They are commonly used in statistics and data analysis due to their prevalence in many natural phenomena and datasets. While they can technically have negative values, they are unlikely to occur and can be considered negligible in most practical applications. Some real-world examples of Gaussians include human height, IQ scores, blood pressure, stock prices, and test scores. However, it is important to note that not all data follows a Gaussian distribution and normality should be checked before making
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mnb96
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Hello,
let's suppose we are given a Gaussian function [tex]f(x)=Ae^{-ax^2}[/tex] (where a,A are real scalars and a is positive)

Is it possible to (dis)prove that the following identity is true only for Gaussians?

[tex]\frac{\partial^2}{\partial x^2} log f(x) = const[/tex]

Thanks!
 
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If you solve the differential equation you end up with
[tex] f(x) = A e^{const\. x^2 + B x}.[/tex]
So provided the constant is negative, you get a gaussian with a shift in the variable.
 

Related to Unique properties of Gaussians

1. What makes Gaussians unique?

Gaussians, also known as Gaussian distributions or normal distributions, are unique because they have a bell-shaped curve and are symmetric around the mean. They also have a well-defined and easily interpretable standard deviation.

2. What are the main properties of Gaussians?

The main properties of Gaussians include their bell-shaped curve, symmetry around the mean, and a well-defined mean and standard deviation. They also have a finite range and the total area under the curve is equal to 1.

3. How are Gaussians used in statistics and data analysis?

Gaussians are commonly used in statistics and data analysis because many natural phenomena and datasets follow a Gaussian distribution. This allows for the use of statistical methods such as the Central Limit Theorem and the 68-95-99.7 rule to make predictions and draw conclusions about the data.

4. Can Gaussians have negative values?

Technically, Gaussians can have negative values, but they are very unlikely to occur. The probability of a negative value occurring in a Gaussian distribution decreases rapidly as the value moves away from the mean. In most practical applications, negative values can be considered negligible.

5. What are some real-world examples of Gaussians?

Some examples of natural phenomena that follow a Gaussian distribution include human height, IQ scores, and blood pressure. In data analysis, stock prices and test scores are often assumed to follow a Gaussian distribution. However, it is important to note that not all data follows a Gaussian distribution and it is important to check for normality before making any assumptions or conclusions.

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