# 03_28 - STAT 410 Examples for Spring 2008 Theorem 3 X X D n...

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Unformatted text preview: STAT 410 Examples for 03/28/2008 Spring 2008 Theorem 3 X X D n → , g is continuous on the support of X & ( ) ( ) X X g g D n → Theorem 4 M X n ( t ) → M X ( t ) for | t | < h & X X D n → Example 12 : Let X n be Binomial ( n , p n = λ / n ). Then M X n ( t ) = n t e n n 1 & & ¡ ¢ £ ¤ ¥ ¦ +- → e λ ( e t – 1 ) as n → ∞ . M X ( t ) = e λ ( e t – 1 ) , where X has a Poisson ( λ ) distribution. & X X D n → ( Poisson approximation to Binomial distribution ). Example 13 : Let X n be χ 2 ( n ). Let Y n = X n / n . Then M Y n ( t ) = E ( e t X n / n ) = M X n ( t / n ) = 2 2 1 1 n n t ¡ ¢ £ ¤ ¥ ¦- → e t as n → ∞ . M X ( t ) = e t , where P ( X = 1 ) = 1. & 1 Y D n → . & 1 Y P n → . Example 14 : Let X n be χ 2 ( n ). Let Z n = ( ) n n n 2 X- M Z n ( t ) = 2 n t e- M X n ( t / n 2 ) = 2 2 2 2 1 1 n n t n t e & & ¡ ¢ £ £ ¤ ¥- ⋅- = 2 2 2 2 n n t n t e e n t- & & & & & ¡ ¢ £ £ £ £ £ ¤ ¥- ,...
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