Statistics for Normal Random Variables

Let the set of independent and identically distributed random variables X₁, Xparameters m_X and s²_X. These ₂, ..., X_m be normally distributed with random variables correspond to a sample of m observations from a N(m_X, s²_X) population. Their mean and sample variance is given by

Because the X_i 's are N(m_X, s²_X) random variables, the random variable X₁ + X₂ + ... + X_m is a N(mm_X, ms²_X) random variable. Therefore

The normality of the X_i's makes it possible for certain statistics involving these random variables to withhave an exact (not approximate) probability distribution a familiar form, Specifically,_i's e.g., having a normal (or related) distribution.

Consider another set of independent and normally distributed random variables Y₁, Y₂, ..., Y_n having parameters m_Y and s²_Y. These random variables correspond to a sample of n observations from a N(m_Y, s²_Y ) population. Their mean and sample variance is given by

Since the Y_i 's are N(m_Y, s²_Y ) random variables, the random variable X₁ + X₂ + ... + X_m is a N(mm_X, ms²_X) random variable. Therefore

and, just like with the X_i's,

If the variances s²_X and s²_Y are unknown but equal to each other, then

where

If the variances s²_X and s²_Y are unknown and not equal to each other, the distribution of has a form that is quite mathematically complex.

The variances s²_X and s²_Y have the following relationship:

All these test statistics have an exact (or approximate, for large sample sizes) probability distribution with the same form as a normal, chi-square, Student t and F distribution. They are used in hypothesis testing and the construction of confidence intervals for significant population parameters.