Computing for any general function whose variable is a gaussian

In summary, we discussed the possibility of computing the resulting gaussian distribution when applying a known function f to a variable X with a known gaussian distribution. However, we found that the resulting distribution may not always be gaussian, such as in cases where f(x)=x^2. We also looked at the expression for calculating the expected value of f(z) when z is a random variable with a gaussian distribution, and found that the correct expression includes a factor of 1/sqrt(2pi).
  • #1
RRraskolnikov
12
0
If I have a variable X whose gaussian distribution is known and let f be a known function, is there a way to compute f(X) (i.e) the resulting gaussian distribution from this? Is the result actually a gaussian distribution?
 
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  • #2
It won't be a gaussian distribution in general. For example if f(x) = x2 then you get what's called the chi-squared distribution with one degree of freedom (k degrees of freedom is adding the square of k gaussians). These are not gaussian (in fact it always has to give a positive number)
 
  • #3
Office_Shredder said:
It won't be a gaussian distribution in general. For example if f(x) = x2 then you get what's called the chi-squared distribution with one degree of freedom (k degrees of freedom is adding the square of k gaussians). These are not gaussian (in fact it always has to give a positive number)

[itex]E(f(z))= \int_{-\inf}^{\inf}{f(z)e^{-\frac{z^2}{2}}}dz[/itex]

What about this above relation? Found it somewhere and it said this is for finding the expected value of f(z) when z is a random variable with gaussian distro.
 
  • #4
That's correct except there should be a 1/sqrt(2pi) in there. In general if you have a probability density function p(x) for a random variable X, then
[tex] E(f(X)) = \int_{-\infty}^{\infty} f(x) p(x) dx [/tex]
In this case your p(x) is the Gaussian density.
 
  • #5
Office_Shredder said:
That's correct except there should be a 1/sqrt(2pi) in there. In general if you have a probability density function p(x) for a random variable X, then
[tex] E(f(X)) = \int_{-\infty}^{\infty} f(x) p(x) dx [/tex]
In this case your p(x) is the Gaussian density.

If you don't mind, can you write down the correct expression with the pi?
 
  • #6
Office_Shredder said:
That's correct except there should be a 1/sqrt(2pi) in there. In general if you have a probability density function p(x) for a random variable X, then
[tex] E(f(X)) = \int_{-\infty}^{\infty} f(x) p(x) dx [/tex]
In this case your p(x) is the Gaussian density.

[itex]E(f(z))= \frac{1}{sqrt(2\pi)}\int_{-\inf}^{\inf}{f(z)e^{-\frac{z^2}{2}}}dz[/itex]


Is this the right expression?

Mod note: Fixed it for you. The LaTeX for infinity is \infty, not \inf. And for the square root, it's \sqrt
$$E(f(z))= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}{f(z)e^{-\frac{z^2}{2}}}dz $$
 
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  • #7
That looks correct. That doesn't say anything about what the distribution of f is, all you are being told is what the expected value is
 

1. What is a gaussian function?

A gaussian function, also known as a normal distribution, is a type of probability distribution that is commonly used in statistics and data analysis. It is characterized by a bell-shaped curve and is often used to model natural phenomena such as human height or test scores.

2. How is computing for a gaussian function different from other types of functions?

Computing for a gaussian function involves specific mathematical algorithms that take into account the unique characteristics of a normal distribution, such as its mean and standard deviation. Other types of functions may have different algorithms or methods for computing their values.

3. Can computing for a gaussian function be done manually?

While it is possible to compute for a gaussian function manually using mathematical formulas, it can be a time-consuming and tedious process. This is why most scientists and statisticians use specialized software or programming languages to perform these calculations.

4. What are some practical applications of computing for gaussian functions?

Gaussian functions are commonly used in various fields such as physics, economics, engineering, and finance. They can be used to model and analyze data, make predictions, and solve complex problems in these areas.

5. Are there any limitations or assumptions when computing for a gaussian function?

Yes, there are certain limitations and assumptions when computing for a gaussian function. Some of these include the assumption that the data follows a normal distribution and that the mean and standard deviation are known. Additionally, the accuracy of the results may depend on the quality and quantity of the data used.

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