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This article explores the distribution of random variables transformed by monotonic functions, specifically focusing on the case where ( Y = g(X) ) and ( X ) follows a Chi-Square distribution with 1 degree of freedom. The concept is illustrated with examples, showing how ( Y ) can also conform to a Chi-Square distribution under certain transformations. We further examine the relation between the Chi-Square and multinomial distributions, particularly in scenarios involving multiple outcomes and large experimental repetitions, where it approximates ( chi^2(K-1) ) degrees of freedom.
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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 ?
2 with 1 degree of freedom (d.f.) What is the distribution of Y ?
f(x) x f(v) v
f(y) y
2(n d.f.) 2(n d.f.) = 2(1 d.f.)* 2(1 d.f.) …* 2(1 d.f.) n times
2 and multinomial distribution Multinomial distribution – K possible outcomes of an experiment probabilities: p1, p2, …, pK, p1+p2+ …+pK=1 N - experiments
For large N Becomes 2(K-1 d.f.)
