P x 1 1 p x k k for any non negative integers x 1 x k

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p x 1 1 · · · p x k k , for any non-negative integers x 1 , · · · , x k s.t. k j =1 x j = n . Jimin Ding, Math WUSTL Math 494 Spring 2018 5 / 11
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Test Statistics and Sampling Distribution Let ( X 1 , · · · , X k ) Multinomial ( p 1 , · · · , p k ) . Define Q = k X j =1 ( X j - np j ) 2 np j . Then Q approximately follows χ 2 distribution with degree of freedom k - 1 , that is Q D -→ χ 2 k - 1 , as n → ∞ . Jimin Ding, Math WUSTL Math 494 Spring 2018 6 / 11
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Test Statistics and Sampling Distribution Let ( X 1 , · · · , X k ) Multinomial ( p 1 , · · · , p k ) . Define Q = k X j =1 ( X j - np j ) 2 np j . Then Q approximately follows χ 2 distribution with degree of freedom k - 1 , that is Q D -→ χ 2 k - 1 , as n → ∞ . In a one-way contingency table, let O j and E j denote the observed count and expected count in the j -th cell, respectively. Then one may write Q = k X j =1 ( O j - E j ) 2 E j . Jimin Ding, Math WUSTL Math 494 Spring 2018 6 / 11
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Example 1: Discrete Uniform Jimin Ding, Math WUSTL Math 494 Spring 2018 7 / 11
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Example 2: Continuous Distribution Jimin Ding, Math WUSTL Math 494 Spring 2018 8 / 11
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Comments I χ 2 is only an approximation, and only works well for large n . In practice, the rule of thumb is : np j 5 , j .
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