Guys I am having trouble with the standard normal distribution.
http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm
We know the X values run from approx $-\infty$ to $+\infty$ but what are the y values??
The normal distribution takes two parameters $\mathcal{N}(\mu, \sigma^2)$ but what is the range of y?
$y>0$ obviously and the "y" will depend on the mean and variance you picked as $y=\frac{\exp(-z^2)}{\sqrt{2\pi\sigma^2}}$. But I have trouble understanding what it means. If I take the S&P500 and I difference the series (SPX-SPX(-1)) the histogram of the returns will have an approximate normal distributions and will list out the number of times I have a return of -1%,-.5%,0%,.5%, 1% , etc throughout the history. So is the "y" of the normal distribution the number of times I have had that x as a value? Should I think of the normal distribution in practical terms the number of times that one point event has occurred? I look at some normal distributions and the Y ranges from 0-4, others I see the y ranging from 0 to 1, as a probability should. I know the area underneath the curve should sum to 1 but shouldnt the y values always be less than 1?
https://statistics.laerd.com/statistical-guides/standard-score.php
Thanks guys!
Best Answer
You may be thinking of the cumulative distribution function, which takes on all values in the interval $(0,1)$. Or else you may be thinking of the (probability) density function $$\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}},$$ the familiar "bell-shaped" curve. This density function is positive, but not necessarily between $0$ and $1$. It reaches a maximum when $x=\mu$. The maximum value (what your post would call the maximum $y$-value) is $\dfrac{1}{\sqrt{2\pi}\sigma}$. The range of the density function is the interval $\left(0,\frac{1}{\sqrt{2\pi}\sigma}\right]$.
In particular, when $\sigma$ is small, the maximum value can be quite large: the density function reaches a sharp high peak. If $\sigma$ is large, the density function, though still characteristically bell-shaped, is flat and low. The area under the density curve, and above the $x$-axis, is always $1$. So if the density function is near $0$ very soon (small variance,) it is intuitively clear that the curve must reach quite high.
Remark: Let $f(x)$ be our probability density function. Then for small $h$, the probability that our random variable lies between $x$ and $x+h$ is approximately $hf(x)$. In that sense, you can pick up a pretty good picture of $f(x)$ if you have a largish number of data points.