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Statistics

Notes Define

Y = X + X
1 2
where X and X are two independent random variables having Chi-square distributions with n
1 2 1
and n degrees of freedom respectively. Prove that Y has a Chi-square distribution with n + n
2 1 2
degrees of freedom.
Solution
The moment generating functions of X and X are
1 2
-n /2
M (t) = (1 – 2t) 1
X 1
-n /2
M (t) = (1 – 2t) 2
X 2
The moment generating function of a sum of independent random variables is just the product
of A random variable X is said to have a Chi-square distribution with n degrees of freedom if its
1
moment generating function is defined for any t  and it is equal to:
2
M (t) = (1 – 2t) -n/2
X
Define
Y = X + X
1 2
where X and X are two independent random variables having Chi-square distributions with n
1 2 1
and n degrees of freedom respectively. Prove that Y has a Chi-square distribution with n + n
2 1 2
degrees of freedom.
Solution
The moment generating functions of X and X are
1 2
-n /2
M (t) = (1 – 2t) 1
X
1
-n /2
M (t) = (1 – 2t) 2
X
2
The moment generating function of a sum of independent random variables is just the product
of their moment generating functions
-n /2 -n /2
M (t) = (1 – 2t) 1 (1 – 2t) 2
Y
= (1 – 2t) –(n + n )/2
1
2
Therefore, M (t) is the moment generating function of a Chi-square random variable with
Y
n + n degrees of freedom. As a consequence, Y has a Chi-square distribution with n + n
1 2 1 2
degrees of freedom.
12.5 Summary
 Moment Generating Function - Defintion
We start this lecture by giving a definition of moment generating function.

Definition: Let X be a random variable. If the expected value:
E[exp(tX)]

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