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17.5 Weighted Least Squares 749

The Student t distribution. Let Z have a standard normal distribution, let W have a xm2
distribution, and let Z and W be independently distributed. Then the random variable

t = Z (17.40)
2W > m

has a Student t distribution with m degrees of freedom, denoted tm. The t ∞ distribution is
the standard normal distribution. (See Exercise 17.15.)

The F distribution. Let W1 and W2 be independent random variables with chi-squared
distributions with respective degrees of freedom n1 and n2. Then the random variable

F = W1 > n1 (17.41)
W2 > n2

has an F distribution with (n1, n2) degrees of freedom. This distribution is denoted Fn1,n2.

The F distribution depends on the numerator degrees of freedom n1 and the denomi-

nator degrees of freedom n2. As number of degrees of freedom in the denominator gets
large, the Fn1,n2 distribution is well approximated by a xn21 distribution, divided by n1. In the
limit, the Fn1, ∞ distribution is the same as the x2n1 distribution, divided by n1; that is, it is the
same as the x2n1>n1 distribution. (See Exercise 17.15.)

A p p e n d i x

17.2 Two Inequalities

This appendix states and proves Chebychev’s inequality and the Cauchy–Schwarz inequality.

Chebychev’s Inequality

Chebychev’s inequality uses the variance of the random variable V to bound the probabil-
ity that V is farther than {d from its mean, where d is a positive constant:

Pr( 0 V - mV 0 Ú d) … var(V) (Chebychev’s inequality). (17.42)
d2

To prove Equation (17.42), let W = V - mV, let f be the p.d.f. of W, and let d be any
positive number. Now