Deriving the Gaussian density probability equation

[CuriousQuazim]

Hey ^^, new here but I already have a question haha

Does anyone here know how the coefficient (x-μ)^2 was derived in the following equation:

σ^3=(1/√2∏)∫(1/σ)*(x-μ)^2*exp((x-μ)^2)/(2σ^2))

I know the general equation for density probability is (1/σ)*exp((x-μ)^2)/(2σ^2))
but in this case I can't quite see how the coefficient came about... any help?

Thanks in advance!

 

[alan2]

Your expression looks wrong to me. Could you check it for accuracy?

 

[mathman]

It looks like it should be σ². The expression is essentially the definition of the variance, the second moment of the distribution centered at the mean.

 

[CuriousQuazim]

Oh I'm sorry that was an error on my part, it is indeed σ^2

σ^2=(1/√2∏)∫(1/σ)*(x-μ)^2*exp((x-μ)^2)/(2σ^2))

Ah thank you so much mathman ^^, that's what I was looking for! I'm studying engineering so sometimes they just throw mathematical equations at us with no explanation ¬_¬.

 

[blue_raver22]

Hello, great question! The Gaussian density probability equation, also known as the normal distribution, is derived from a combination of the central limit theorem and the properties of the exponential function. The coefficient (x-μ)^2 in the equation represents the squared difference between a data point x and the mean μ.

To understand how this coefficient was derived, let's first look at the general equation for density probability that you mentioned: (1/σ)*exp((x-μ)^2)/(2σ^2)). This equation represents the probability density function for a normal distribution, where σ is the standard deviation and μ is the mean.

Now, the central limit theorem states that the sum of a large number of independent and identically distributed random variables will follow a normal distribution. This means that if we take a large number of data points and calculate their mean, the resulting distribution will be a normal distribution with mean μ and standard deviation σ.

To find the probability of a specific data point x occurring within this distribution, we use the properties of the exponential function. The exponential function has a form of e^x, where e is the base of the natural logarithm. When we take the natural logarithm of this function, we get ln(e^x) = x. This means that the natural logarithm of the exponential function is equal to the exponent itself.

Now, if we apply this property to our general equation for density probability, we get ln((1/σ)*exp((x-μ)^2)/(2σ^2)) = ln((1/σ)*exp(-(x-μ)^2/(2σ^2))). This simplified equation is the same as our original equation, but now we can see that the exponent is equal to -(x-μ)^2/(2σ^2). This is where the coefficient (x-μ)^2 comes from.

In summary, the coefficient (x-μ)^2 in the Gaussian density probability equation represents the squared difference between a data point x and the mean μ. This is derived from the properties of the exponential function and the central limit theorem. I hope this helps! Let me know if you have any further questions.