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4. Functions of Random Variables and Sampling Distribution 209

In other words, has the same representation as that of W with
ν = n – 1. Hence, we can claim that

A standardized variable such as is widely used in the sequel
2
when the population mean µ and variance σ are both unknown.
W. S. Gosset was a pioneer in the development of statistical methods for
design and analysis of experiments. He is perhaps better known under the
pseudonym “Student” than under his own name. In most of his papers, he
preferred to use the pseudonym “Student” instead of his given name. His
path-breaking 1908 paper gave the foundation of this t-distribution. !
Example 4.5.2 The Two-Sample Problem: Suppose that the random vari-
2
ables X , ..., X are iid N(µ , σ ), i = 1, 2, and that the X ’s are independent
1j
i1
i
ini
of the X ’s. With n ≥ 2, let us denote
2j i
for i = 1, 2.

is called the pooled sample variance.

Now, 2, and these are also independent. Us-
ing the reproductive property of independent Chi-squares (Theorem 4.3.2,
part (iii)) we claim that has a Chi-square dis-
tribution with (n + n – 2) degrees of freedom. Also, and are
2
1
independent. Along the lines of the Example 4.5.1, we can claim that

This two-sample Student’s t distribution is widely used in the statistical litera-
ture. !

4.5.2 The F Distribution

Definition 4.5.2 Let X, Y be independent Chi-square random variables dis-
tributed respectively with í and í degrees of freedom. Then, the random
2
1
variable U = (X/ ν ) ÷ (Y/ ν ) is said to have the F distribution with degrees
1
2
of freedom ν , ν , in that order.
1 2