What are the properties of a normal distribution? Why there are an infinite number of normal distributions? Do you want to assume that your sample data represent a population distribution? Solution Properties: Function f(x) is unimodal and symmetric around the point x = µ, which is at the same time the mode, the median and the mean of the distribution.[8] The inflection points of the curve occur one standard deviation away from the mean (i.e., at x = µ - s and x = µ + s).[8] Function f(x) is log-concave.[8] The standard normal density ?(x) is an eigenfunction of the Fourier transform. The function is supersmooth of order 2, implying that it is infinitely differentiable.[9] The first derivative of ?(x) is ?\'(x) = -x·?(x); the second derivative is ?\'\'(x) = (x2 - 1)?(x). More generally, the n-th derivative is given by ?(n)(x) = (-1)nHn(x)?(x), where Hn is the Hermite polynomial of order n.[10] When s2 = 0, the density function doesn\'t exist. However a generalized function that defines a measure on the real line, and it can be used to calculate, for example, expected value is where d(x) is the Dirac delta function which is equal to infinity at x = 0 and is zero elsewhere. The normal distribution is a continuous distribution defined by two parameters, the mean and the variance. since the variance can be any positive value and the mean can be any real value there are infinitely many different values for the mean and for the variance..

1. What are the properties of a normal distribution? Why there are an infinite number of normal
distributions? Do you want to assume that your sample data represent a population distribution?
Solution
Properties: Function f(x) is unimodal and symmetric around the point x = µ, which
is at the same time the mode, the median and the mean of the distribution.[8] The inflection
points of the curve occur one standard deviation away from the mean (i.e., at x = µ - s and x = µ
+ s).[8] Function f(x) is log-concave.[8] The standard normal density ?(x) is an eigenfunction of
the Fourier transform. The function is supersmooth of order 2, implying that it is infinitely
differentiable.[9] The first derivative of ?(x) is ?'(x) = -x·?(x); the second derivative is ?''(x) =
(x2 - 1)?(x). More generally, the n-th derivative is given by ?(n)(x) = (-1)nHn(x)?(x), where Hn
is the Hermite polynomial of order n.[10] When s2 = 0, the density function doesn't exist.
However a generalized function that defines a measure on the real line, and it can be used to
calculate, for example, expected value is where d(x) is the Dirac delta function which is equal to
infinity at x = 0 and is zero elsewhere. The normal distribution is a continuous distribution
defined by two parameters, the mean and the variance. since the variance can be any positive
value and the mean can be any real value there are infinitely many different values for the mean
and for the variance.