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The probability density must be scaled by 1 / σ {\displaystyle 1/\sigma } so that the integral is still1.
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The parameter μ {\displaystyle \mu } is the mean or expectation of the distribution (and also its median and mode), while the parameter σ {\displaystyle \sigma } is its standard deviation. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Conversely, if X {\displaystyle X} is a normal deviate with parameters μ {\displaystyle \mu } and σ 2 {\displaystyle \sigma If Z {\displaystyle Z} is a standard normal deviate, then X = σ Z + μ {\displaystyle X=\sigma Z+\mu } will have a normal distribution with expected value μ {\displaystyle \mu } and standard deviation σ {\displaystyle \sigma } .
Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions). Although the density above is most commonly known as the standard normal, a few authors have used that term to describe other versions of the normal distribution. The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution. In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable.
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The density φ ( z ) {\displaystyle \varphi (z)} has its peak 1 / 2 π {\displaystyle 1/{\sqrt {2\pi }}} at z = 0 {\displaystyle z=0} and inflection points at z = + 1 {\displaystyle z=+1} and z = − 1 {\displaystyle z=-1} . The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution.