# B Probability density of a normal distribution

1. Jan 6, 2017

If the normalized probability density of the normal distribution is $p(x) = \frac {1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$, then if $\sigma = 0.0001$ and in the special case $x = \mu$, wouldn't the probability density at this point, $p(\mu)$, exceed 1 since it is equal to $p(\mu) = \frac {1}{\sqrt{2\pi}0.0001} > 1$? Wouldn't this mean it is not normalized?

2. Jan 6, 2017

### FactChecker

No. The density function can get huge as long as its integral is equal to 1. So the density function can get very large for a short range of X.

3. Jan 6, 2017

Okay, just wanted to ensure I understood that. Thank you. So in general, the value for $p(x)$ always varies from 0 to $\frac{1}{\sqrt{2\pi}\sigma}$?

4. Jan 6, 2017

### Staff: Mentor

Yes.

Other probability distributions can have even higher densities - as long as they are in a small range. Only the integral is important.

5. Jan 7, 2017

### Stephen Tashi

As an analogy, a 1 kg rock with a volume of 300 cc could have a density of 1.5 kg/ 300 cc at a particular location within the rock.