Download The Chi-Square Distribution

2.3.
The Chi-Square Distribution
One of the most important special cases of the gamma distribution is the chi-square
distribution because the sum of the squares of independent normal random variables with
mean zero and standard deviation one has a chi-square distribution. This section collects
some basic properties of chi-square random variables, all of which are well known; see
Hogg and Tanis [6].
A random variable X has a chi-square distribution with n degrees of freedom if it is a
gamma random variable with parameters m = n/2 and  = 2, i.e X ~ (n/2,2). Therefore,
its probability density function (pdf) has the form
 2  (n/2)

0
t(n/2)-1e-t/2
n/2
(1)
f(t) = f(t; n) =
if t > 0
if t < 0
In this case we shall say X is a chi-square random variable with n degrees of freedom and
write X ~ (n). Usually n is assumed to be an integer, but we only assume n > 0.
Proposition 1. If X has a gamma distribution with parameters m and  then 2X/ has a
chi-square distribution with 2m degrees of freedom.
Proof. By Proposition 5 in section 2.2 the random variable X has a gamma distribution
with parameters m and 2, i.e X ~ (m,2) = ((2m)/2,2). The proposition follows from
this. 
Proposition 2. If X has a chi-square distribution with n degrees of freedom, then the
mean of X is X = E(X) = n. If Y/ has a chi-square distribution with n degrees of
freedom, then the mean of Y is Y = E(Y) = n.
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Proof. Since X ~ (n/2,2) it follows from Proposition 2 of section 2.2 that
X = (n/2)(2) = n. One has Y/ = X where X has a chi-square distribution with n degrees
of freedom. Therefore E(Y) = E(X) = E(X) = n. 
Proposition 3. If X has a chi-square distribution with n degrees of freedom, then the
variance of X is X2 = E((X - X)2) = 2n. If Y/ has a chi-square distribution with n
degrees of freedom, then the variance of Y is Y2 = 2n2.
Proof. Since X ~ (n/2,2) it follows from Proposition 3 of section 2.2 that
X2 = (n/2)(22) = 2n. One has Y/ = X where X has a chi-square distribution with n
degrees of freedom. Therefore Y2 = 2X2 = 2n2. 
Proposition 4. If f(t) is given by (1) then for t > 0 one has
f(t) has a single local maximum at t = n - 2 if m > 2.
f(t) is strictly decreasing for t > 0 if m  2
Proof. Since X ~ (n/2,2) this follows from Proposition 4 of section 2.2. 
Proposition 5. If X and Y are independent chi-square random variables with n and p
degrees of freedom respectively, then X + Y is a chi-square random variable with n + p
degrees of freedom.
Proof. Since X ~ (n/2,2) and Y ~ (p/2,2), it follows from Proposition 5 of section 2.2
that X + Y ~ ((n+p)/2,2). The proposition follows from this. 
Proposition 6. If X has a chi-square distribution with n degrees of freedom, then the
Laplace transform L(s) and moment generating function M(r) of X are given by
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1
L(s) = (1 + 2s)n/2
1
M(r) = (1 - 2r)n/2
Proof. Since X ~ (n/2,2) this follows from Proposition 6 of section 2.2. 
Proposition 7. Let Z1, …, Zn be independent normal random variables with mean zero
and standard deviation one and let S = Z12 + … + Zn2. Then S has a chi square
distribution with n degrees of freedom.
Proof. First consider the case n = 1, i.e. S =Z2 where Z is a normal random variable with
mean zero and standard deviation one. Let F(s) = Pr{S  s} be the distribution function
of S. Then for s > 0 one has
s
F(s) = Pr{U2  s} = Pr{ - s  U  s } = 
 g(u) du
- s
where g(u) is the density function of Z. Therefore, the density function of S is
(19)
dF
e-s/2
s1/2-1e-s/2
1
-1
1
f(s) = ds = g( s)   - g( s)   = g( s)   =
= 1/2
2  (1/2)
2 s
2 s
 s
2s
This is a gamma random variable with parameters m = 1/2 and  = 2.so the result is true
for n = 1. The case of general n follows from Proposition 5. 
Corollary 8. Let V1, …, Vn be independent normal random variables with mean zero and
standard deviation  and let W = V12 + … + Vn2. Then W/2 has a chi-square distribution
with n degrees of freedom and W is a gamma random variable with parameters n/2 and
22.
Proof. Vj = Zj where the Zj are independent normal random variables with mean zero
and standard deviation 1. So W = 2S where S = Z12 + … + Zn2. By Proposition 7, S has
a chi-square distribution with n degrees of freedom and the result follows. 
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Corollary 9. Let U1, …, Un be independent normal random variables with mean  and
standard deviation  and let W = (U1 - )2 + … + (Un - )2. Then W/2 has a chi-square
distribution with n degrees of freedom.
Proof. W = V12 + … + Vn2 where Vj = Uj - . The Vj are independent normal random
variables with mean zero and standard deviation . So the result follows from Corollary
8. 
Theorem 10. Let U1, …, Un be independent normal random variables with the same
_
mean and standard deviation  and let U = (U1 + … + Un)/n and
_
_
W = (U1 - U)2 + … + (Un - U)2. Then W has a chi square distribution with n – 1 degrees
of freedom and W is a gamma random variable with parameters (n-1)/2 and 22.
The proof of this is more involved; see Rao [11, p. 147].
Let X have a chi-square distribution with n degrees of freedom and let
t
t
(n/2)-1 -s/2
s
e
G(t) = G(t;n) = 
ds
 f(s;n) ds = 
2n/2 (n/2)

0
0
be its cummulative distribution function and let

m-1 -s
m(t) = 
 s e ds
t
be the upper incomplete gamma function and
t
 sm-1e-s ds
m(t) = 
0
be the lower incomplete gamma function.
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Theorem 11.
 (t/2)
G(t;m,) = n/2(n/2)
Proof. Since X ~ (n/2,2) this follows from Proposition 7 of section 2.2. 
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