Chi - square

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Chi - square
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Distributions of functions of r.v.s
X has a probability density function f(x) We
define Y g(X), where g(.) is monotonic
function What is the distribution of Y ?
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Examples
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?2 with 1 degree of freedom (d.f.)
What is the distribution of Y ?
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We introduce V X
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f(x)
x
f(v)
v
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Y ?2(1 d.f.)
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f(y)
y
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?2(n d.f.)

• ?2(n d.f.) ?2(1 d.f.) ?2(1 d.f.) ?2(1 d.f.)

n times
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Y ?2(n d.f.)
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?2 and multinomial distribution

• Multinomial distribution K possible outcomes
  of an experiment
• probabilities p1, p2, , pK,
• p1p2 pK1
• N - experiments

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For large N
Becomes ?2(K-1 d.f.)