The discussion centers on the properties of the probability density function of the normal distribution, particularly addressing the implications of having a very small standard deviation (σ) and whether this leads to a violation of normalization when evaluating the density at the mean (μ).
Participants generally agree that the normalization condition is satisfied as long as the integral of the probability density function equals 1, but there is disagreement regarding the implications of high density values at specific points.
The discussion does not resolve the implications of high density values in relation to normalization, and the assumptions regarding the behavior of probability densities in small ranges remain unexamined.
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.TheCanadian said: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?
FactChecker said: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.