Problems with Gaussian distribution

In summary, the conversation is about using integration to find the value of A in a Gaussian distribution. The person is struggling to figure out the correct value for alpha in the integration formula and is seeking help. Another person suggests doing a change of variables to make the integration easier.
  • #1
cooper607
49
0

Homework Statement



consider this Gaussian distribution
p(x)=Ae^-(a(x-b)^2)

Homework Equations



use integration p(x)dx=1 to find out the value of A

The Attempt at a Solution



hi, i know about the gaussian distribution formula integration e^-alpha*x^2 = sqrt(pi/alpha)

now for this integration i just could not figure out what the alpha should be. as if i want to get the moderate Gaussian form i ended up with e^-x^2(a-2ba/x+b^2*a/x^2)

as i could not get rid of x in my alpha term , can i still integrate it with the gaussian formula?
if not , then how can i fix my alpha here containing no x terms?
regards
 
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  • #2
cooper607 said:

Homework Statement



consider this Gaussian distribution
p(x)=Ae^-(a(x-b)^2)

Homework Equations



use integration p(x)dx=1 to find out the value of A

The Attempt at a Solution



hi, i know about the gaussian distribution formula integration e^-alpha*x^2 = sqrt(pi/alpha)

now for this integration i just could not figure out what the alpha should be. as if i want to get the moderate Gaussian form i ended up with e^-x^2(a-2ba/x+b^2*a/x^2)

as i could not get rid of x in my alpha term , can i still integrate it with the gaussian formula?
if not , then how can i fix my alpha here containing no x terms?
regards

Do a change of variables, u=x-b. du=dx. Now integrate du instead of dx.
 
  • #3
wow! that helps.. thanks a lot
 

1. What is a Gaussian distribution?

A Gaussian distribution, also known as a normal distribution, is a probability distribution that is commonly used to model the distribution of a continuous variable. It is characterized by a symmetrical bell-shaped curve, with most data points falling near the mean and fewer data points in the tails.

2. What are some common problems with Gaussian distribution?

Some common problems with Gaussian distribution include outliers, skewness, and kurtosis. Outliers are extreme values that can significantly impact the mean and standard deviation of the data, potentially resulting in a non-normal distribution. Skewness refers to an asymmetry in the distribution, where one tail is longer than the other. Kurtosis measures the peakedness or flatness of the distribution, with a high kurtosis indicating a sharper peak and a low kurtosis indicating a flatter peak.

3. How do you detect non-normality in a Gaussian distribution?

There are a few ways to detect non-normality in a Gaussian distribution. One way is to visually inspect the histogram of the data and look for any obvious deviations from a bell-shaped curve. Another way is to use a statistical test, such as the Shapiro-Wilk test, to determine if the data significantly deviates from a normal distribution. Additionally, you can use measures of skewness and kurtosis to assess the shape of the distribution.

4. What are the implications of non-normality in a Gaussian distribution?

Non-normality in a Gaussian distribution can have several implications. It may affect the accuracy of statistical analyses that assume a normal distribution, such as t-tests and ANOVAs. In these cases, it is important to use alternative methods that do not rely on normality assumptions. Non-normality can also impact the interpretation of results, as it may indicate that the underlying data is not representative of the population.

5. Can non-normal data be transformed to fit a Gaussian distribution?

Yes, non-normal data can often be transformed to fit a Gaussian distribution. This can be done through various mathematical transformations, such as logarithmic or square root transformations. However, it is important to note that transforming data can also alter the relationships between variables and may not always be appropriate. It is important to carefully consider the purpose of the analysis and the assumptions of the chosen statistical method before deciding to transform the data.

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