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 ?

We introduce V = | X |

f(x) x f(v) v

Y ~ 2(1 d.f.)

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

Y ~ 2(n d.f.)

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.)