Introduction to Probability and Statistics for Engineers and Scientists

190 Chapter 5: Special Random Variables

0.4

0.3

0.2

0.1

− 3 − 2 −10 1 2 3

t-density with 5 degrees of freedom

Standard normal density

FIGURE 5.15 Comparing standard normal density with the density of T 5.

The mean and variance ofTncan be shown to equal

E[Tn]=0, n> 1

Var(Tn)=

n

n− 2

, n> 2

Thus the variance ofTndecreases to 1 — the variance of a standard normal random
variable — asnincreases to∞. Forα,0<α<1, lettα,nbe such that

P{Tn≥tα,n}=α

It follows from the symmetry about zero of thet-density function that−Tnhas the same
distribution asTn, and so

α=P{−Tn≥tα,n}

=P{Tn≤−tα,n}

= 1 −P{Tn>−tα,n}

Therefore,

P{Tn≥−tα,n}= 1 −α

leading to the conclusion that

−tα,n=t 1 −α,n

which is illustrated in Figure 5.16.
The values oftα,nfor a variety of values ofnandαhave been tabulated in Table A3
in the Appendix. In addition, Programs 5.8.2a and 5.8.2b on the text disk compute the
t-distribution function and the valuestα,n, respectively.