The following corollary is now obtained by combining Theorem 5.6.1 and Theorem 5.5.2.
Corollary 5.6.1. If the random variance _i(i=1,...,k), and if a1,...,ak and b are constants for which at least one of the values a1,...,ak is different from 0, then the variable a1X1+...+ak k+b has a normal distribution with mean a1U1+..+akUk+b and variance ___+...+_____.
The distribution if the sample mean for a random sample from a normal distribution can now be derived easily.