5.8Distributions Arising from the Normal 187

*5.8.1.1 THE RELATION BETWEEN CHI-SQUARE AND GAMMA RANDOM VARIABLES

Let us compute the moment generating function of a chi-square random variable withn
degrees of freedom. To begin, we have, whenn=1, that

E[etX]=E[etZ

2

]whereZ∼N(0, 1) (5.8.2)

=

∫∞

−∞

etx

2

fZ(x)dx

=

1

√

2 π

∫∞

−∞

etx

2

e−x

(^2) /2
dx

1
√
2 π
∫∞
−∞
e−x
(^2) (1− 2 t)/2
dx

1
√
2 π
∫∞
−∞
e−x
(^2) /2σ ̄ 2
dx whereσ ̄^2 =(1− 2 t)−^1
=(1− 2 t)−1/2
1
√
2 πσ ̄
∫∞
−∞
e−x
(^2) /2σ ̄ 2
dx
=(1− 2 t)−1/2
where the last equality follows since the integral of the normal (0,σ ̄^2 ) density equals 1.
Hence, in the general case ofndegrees of freedom
E[etX]=E
[
et
∑n
i= 1 Zi^2
]
=E
[n
∏
i= 1
etZ
i^2
]

∏n
i= 1
E[etZ
i^2
] by independence of theZi
=(1− 2 t)−n/2 from Equation 5.8.2
However, we recognize[1/(1− 2 t)]n/2as being the moment generating function of a
gamma random variable with parameters (n/2, 1/2). Hence, by the uniqueness of moment
generating functions, it follows that these two distributions — chi-square withndegrees
of freedom and gamma with parametersn/2 and 1/2 — are identical, and thus we can

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