Sums of Gamma Random Variables - University of Michigan
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
(1) f(t) = f(t; n) =
In this case we shall say X is a chi-square random variable with n degrees of freedom and write X ~ (2(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(.
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
L(s) =
M(r) =
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
F(s) = Pr{U2 ( s} = Pr{ - } =
where g(u) is the density function of Z. Therefore, the density function of S is
(19) f(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. (
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 = (U1 + … + Un)/n and 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
G(t) = G(t;n) = =
be its cummulative distribution function and let
(m(t) =
be the upper incomplete gamma function and
(m(t) =
be the lower incomplete gamma function.
Theorem 11.
G(t;m,() =
Proof. Since X ~ ((n/2,2) this follows from Proposition 7 of section 2.2. (
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