Bsrgvty

I have been studying the probability density function...

$$frac1sigma sqrt2 pie^frac(-(x - mu ))^22sigma ^2$$

For now I remove the constant, and using the following proof, I prove that...

$$int_-infty^inftye^frac-x^22 = sqrt2 pi $$

The way I interpret this is that the area under the gaussian distribution is $sqrt2 pi $. But I am still having a hard time figuring out what the constant is doing. It seems to divide by the area itself and by $sigma$ as well. Why is this done?

If you consider every possible outcome of some event you should expect the probability of it happening to be $1$, not $sqrt2pi$ so the constant scales the distribution to conform with the normal convention of ascribing a probability between zero and one.

It is doing that, but observe that you are also stretching in the horizontal direction by the same factor (in the exponential). Say if $sigma>1$ you are decreasing your area by a factor $sigma$ but you are increasing it by the same factor because you replace $x$ by $x/sigma$ (the shift does not change the area of course)

As you have correctly stated, the p.d.f. of the normal distribution is given by $$f(xmidmu,sigma^2)=frac1sigmasqrt2piexpleft(-frac12left(fracx-musigmaright)^2right)$$ where the parameter space is $mathitTheta=(mu,sigma^2)inBbb R^2:sigma^2>0$. This is essentially saying that the mean is a value on the real line, and the variance is one on the positive real line.

Now consider the simple case where $mu=0$ and $sigma^2=1$. Then the standard normal distribution has p.d.f. $$f(x)=frac1sqrt2piexpleft(-frac12x^2right).$$ If we integrate this in the interval $(-infty,infty)$, we will get $1$. This is by definition always the case as for all $xinmathit X$ (in this instance $mathit X=Bbb R$), $$int_mathit Xf(x),dx=1.$$ That is, the sum of all the probabilities of $x$ being in each region in $mathit X$ is $1$. In fact, the constant that makes this happen is so important in statistics (especially Bayesian statistics) that it is given a name: the normalising constant.

A further example is the Beta distribution, with p.d.f. $$f(xmidalpha,beta)=fracx^alpha-1(1-x)^beta-1text B(alpha,beta)$$ where $1/text B(alpha,beta)$ is the normalising constant.