Furthermore, the four random variables … and … are independent. This result is implied by the following two facts: (i) If one random vector is a function of only, then these two vectors must be independent. (ii) By theorem 7.3.1, … and … are independent, and … and … are also independent, It follows that the two random variables … and ... are independent. The random variable … has a standard normal distribution when … and the random variable … has a … distribution with … degrees of freedom for all values of … , and … . The statistic … can now be represented in the form