Gaussian normal distribution curve

In summary, the conversation discusses the search for a mathematical equation similar to the Gaussian normal distribution curve, but terminating at a finite x = X instead of infinity. The suggested equations include a Gaussian function for a finite interval, a continuous function with a raised cosine, and a function using the Heaviside function. However, it is noted that all of these functions involve piecewise definitions and it is unlikely to find a single mathematical relationship that can cover the entire range of x.
  • #1
touqra
287
0
I am looking for a mathematical equation which is similar to the Gaussian normal distribution curve, but I need one which terminates at a finite x = X and not at infinity, ie, [tex]f(x \geq X) = 0[/tex], but, [tex] f(x \leq X) = [/tex]a function which has a Gaussian shape-like curve.
Is there one such as this that you know or in mathematics literatures?
 
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  • #2
define the Gaussian for any finite interval
f(x)
= gaussian for a<x<b
= 0 otherwise

-- AI
 
  • #3
Or consider:
[tex]f(x)=Ae^{-x^2}-b[/tex]
for [itex]|x|<\sqrt{\ln(A/b)}[/itex] and 0 elswhere.
That one is even continuous.
 
  • #4
Or even just a raised cosine, 1 + cos(x) : -pi < x < pi .
 
  • #5
Is there a mathematical function which is not piece-wise and can be defined for the whole range of x (just by one mathematical relationship)?
 
  • #6
Probably not since you are splitting the range up in your own requirements, besides, that is purely a superficial issue for you. If you know what a heaviside function is then, for k some positive real number

H(x+k)(1-H(x-k))f(x)

for some suitably shaped and nomalized function would do and would appear to be a nice single line wouldn't it? Of course

H(y) is defined piecewise.
 

What is the Gaussian normal distribution curve?

The Gaussian normal distribution curve, also known as the bell curve, is a symmetric probability distribution that is often used to model the distribution of continuous variables in natural and social sciences. It is characterized by its bell shape, with the majority of the data falling within one standard deviation of the mean.

What are the properties of the Gaussian normal distribution curve?

The Gaussian normal distribution curve has several key properties, including symmetry, unimodality (having one peak), and the majority of the data falling within one, two, and three standard deviations from the mean (68%, 95%, and 99.7%, respectively). It also has a mean and standard deviation that can be used to describe its central tendency and spread.

How is the Gaussian normal distribution curve calculated?

The Gaussian normal distribution curve is calculated using a mathematical formula that takes into account the mean and standard deviation of the data. The formula, also known as the probability density function, is f(x) = (1/σ√(2π)) * e^(-(x-μ)^2/2σ^2), where μ is the mean and σ is the standard deviation.

What is the purpose of using the Gaussian normal distribution curve?

The Gaussian normal distribution curve is used to understand and analyze data that follows a normal distribution. It allows researchers to make predictions and draw conclusions based on the probability of certain outcomes occurring within a given range of values. It is also used in statistical tests to determine the significance of results.

How is the Gaussian normal distribution curve different from other types of distributions?

The Gaussian normal distribution curve is different from other types of distributions in that it is symmetric and unimodal. Other distributions, such as the skewed distribution or the bimodal distribution, have different shapes and may not follow the same mathematical formula. The Gaussian normal distribution curve is also commonly used because of its simplicity and its ability to approximate many natural phenomena.

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