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Chi - square

Chi - square. 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 ?. Examples.  2 with 1 degree of freedom (d.f.). What is the distribution of Y ?.

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Chi - square

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  1. Chi - square

  2. 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 ?

  3. Examples

  4. 2 with 1 degree of freedom (d.f.) What is the distribution of Y ?

  5. We introduce V = | X |

  6. f(x) x f(v) v

  7. Y ~ 2(1 d.f.)

  8. f(y) y

  9. 2(n d.f.) 2(n d.f.) = 2(1 d.f.)* 2(1 d.f.) …* 2(1 d.f.) n times

  10. Y ~ 2(n d.f.)

  11. 2 and multinomial distribution Multinomial distribution – K possible outcomes of an experiment probabilities: p1, p2, …, pK, p1+p2+ …+pK=1 N - experiments

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

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