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The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z ...
The normal-exponential-gamma distribution; The normal-inverse Gaussian distribution; The Pearson Type IV distribution (see Pearson distributions) The Quantile-parameterized distributions, which are highly shape-flexible and can be parameterized with data using linear least squares. The skew normal distribution
Frequency distribution: a table that displays the frequency of various outcomes in a sample. Relative frequency distribution: a frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample (i.e. sample size). Categorical distribution: for discrete random variables with a finite set of values.
An example of such distributions could be a mix of discrete and continuous distributions—for example, a random variable that is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2.
In the case of normalization of scores in educational assessment, there may be an intention to align distributions to a normal distribution. A different approach to normalization of probability distributions is quantile normalization, where the quantiles of the different measures are brought into alignment.
The fact that two random variables and both have a normal distribution does not imply that the pair (,) has a joint normal distribution. A simple example is one in which X has a normal distribution with expected value 0 and variance 1, and = if | | > and = if | | <, where >. There are similar counterexamples for more than two random variables.
The empirical distribution of the data (the histogram) should be bell-shaped and resemble the normal distribution. This might be difficult to see if the sample is small. In this case one might proceed by regressing the data against the quantiles of a normal distribution with the same mean and variance as the sample. Lack of fit to the ...
The probability distribution for the sex of the sample can now be determined: = = (()), where = and = are the parameters of normal distribution which have been previously determined from the training set.