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
(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 ~ (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 = 2n2.
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
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 ~ (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 = 2n2.
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