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 ?

4. Examples

5. 2 with 1 degree of freedom (d.f.)
What is the distribution of Y ?

6. We introduce V = | X |

7. f(x) x
f(v) v

9. Y ~ 2(1 d.f.)

10. f(y) y

11. 2(n d.f.)
2(n d.f.) = 2(1 d.f.)* 2(1 d.f.) …* 2(1 d.f.)
n times

12. Y ~ 2(n d.f.)

13. 2 and multinomial distribution
Multinomial distribution – K possible outcomes of an experiment
probabilities: p1, p2, …, pK,
p1+p2+ …+pK=1
N - experiments

14. For large N
Becomes 2(K-1 d.f.)