Explanation Given information: f n ( x ) = c n x ( n − 2 ) / 2 e − x / 2 , x ≥ 0 , where c n is an appropriate constant. Explanation: The chi-square density function with n degrees of freedom is f n ( x ) = c n x ( n − 2 ) / 2 e − x / 2 , x ≥ 0 , where c n is an appropriate constant. The chi-square density function with n = 2 degrees of freedom is f 2 ( x ) = c 2 x ( 2 − 2 ) / 2 e − x / 2 ⇒ f 2 ( x ) = c 2 e − x / 2 Since it is probability density function, ∫ 0 ∞ c 2 e − x / 2 d x = 1 ⇒ c 2 lim b → ∞ ∫ 0 b e − x / 2 d x = 1 ⇒ c 2 lim b → ∞ [ − 2 e − x / 2 | 0 b ] = 1 ⇒ c 2 lim b → ∞ [ − 2 e − b / 2 + 2 e − 0 / 2 ] = 1 ⇒ c 2 ⋅ 2 = 1 Thus, c 2 = 1 2 . The chi-square density function with n = 4 degrees of freedom is f 4 ( x ) = c 4 x ( 4 − 2 ) / 2 e − x / 2 ⇒ f 4 ( x ) = c 4 x e − x / 2 Since it is probability density function, ⇒ ∫ 0 ∞ c 4 x e − x / 2 d x = 1 ⇒ c 4 lim b → 0 ∫ 0 b x e − x / 2 d x = 1 ……(1) Now ∫ 0 b x e − x / 2 d x using integration by parts: The integration by parts rule is ∫ f ( x ) g ( x ) = f ( x ) G ( x ) − ∫ f ' ( x ) G ( x ) d x

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ISBN: 9780137590810

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar, William Edward Tavernetti

Publisher: PEARSON

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The values for c 2 and c 4 , where f n ( x ) = c n x ( n − 2 ) / 2 e − x / 2 , x ≥ 0

Solution Summary: The author explains the chi-square density function with n=2 degrees of freedom.

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The values for c2 and c4, where fn(x)=cnx(n−2)/2e−x/2, x≥0

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