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The Beta Distribution
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A random variable is said to have the beta distribution, or to be beta distributed, if the density function is xa 1(1 x)b u B(a, b) 0
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where B( , ) is the beta function (see Appendix A). In view of the relation (9), Appendix A, between the beta and gamma functions, the beta distribution can also be defined by the density function
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(a b) xa 1(1 (a) (b) 0
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otherwise
where ,
are positive. The mean and variance are m a a b , s2 (a ab b)2 (a b 1) (36)
1 there is a unique mode at the value xmode a a b 1 2 (37)
CHAPTER 4 Special Probability Distributions
The Chi-Square Distribution
Let X1, X2, . . . , Xv be v independent normally distributed random variables with mean zero and variance 1. Consider the random variable x2 where
2 2 X1
X2 2
c 0,
2 Xv
(38)
is called chi square. Then we can show that for x P(x2 x)
1 u(v>2) 1 e 2v>2 (v>2) 30
(39)
and P( 2 x) 0 for x 0. The distribution defined by (39) is called the chi-square distribution, and v is called the number of degrees of freedom. The distribution defined by (39) has corresponding density function given by 1 x(v>2) 1 e 2v>2 (v>2) u0
0 (40) 0 v>2 2. (41)
f (x)
It is seen that the chi-square distribution is a special case of the gamma distribution with Therefore, v,
M(t)
v>2,
2i )
For large v(v 30), we can show that !2x2 !2v 1 is very nearly normally distributed with mean 0 and variance 1. Three theorems that will be useful in later work are as follows: Theorem 4-3 Let X1, X2, . . . , Xv be independent normally distributed random variables with mean 0 and varic X 2 is chi-square distributed with v degrees of freedom. ance 1. Then x2 X 2 X 2 1 2 v Theorem 4-4 Let U1, U2, . . . , Uk be independent random variables that are chi-square distributed with v1, c U is chiv2, . . . , vk degrees of freedom, respectively. Then their sum W U1 U2 k c v degrees of freedom. square distributed with v1 v2 k Theorem 4-5 Let V1 and V2 be independent random variables. Suppose that V1 is chi-square distributed with v1 degrees of freedom while V V1 V2 is chi-square distributed with v degrees of freedom, where v v1. Then V2 is chi-square distributed with v v1 degrees of freedom. In connection with the chi-square distribution, the t distribution (below), the F distribution (page 116), and others, it is common in statistical work to use the same symbol for both the random variable and a value of that random variable. Therefore, percentile values of the chi-square distribution for v degrees of freedom are denoted by x2 , or briefly x2 if v is understood, and not by xp,v or xp. (See Appendix E.) This is an amp,v p biguous notation, and the reader should use care with it, especially when changing variables in density functions.
Student s t Distribution
If a random variable has the density function a f (t) v 2 1 b
v 2vp a b 2
t2 vb
(v 1)>2
(42)
it is said to have Student s t distribution, briefly the t distribution, with v degrees of freedom. If v is large (v 30), the graph of f (t) closely approximates the standard normal curve as indicated in Fig. 4-2. Percentile
CHAPTER 4 Special Probability Distributions
Fig. 4-2
values of the t distribution for v degrees of freedom are denoted by tp,v or briefly tp if v is understood. For a tp; for example, table giving such values, see Appendix D. Since the t distribution is symmetrical, t1 p t0.95. t0.5 For the t distribution we have 0 and s2 v v 2 (v 2). (43)
The following theorem is important in later work. Theorem 4-6 Let Y and Z be independent random variables, where Y is normally distributed with mean 0 and variance 1 while Z is chi-square distributed with v degrees of freedom. Then the random variable T Y 2Z>v (44)
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