Probability density functions

Rules for a valid Probability Density Function (PDF):

1. $f(x)\ge 0$ for all x
2. ${\int }_{-\infty }^{\infty }f(x)dx=1$
3. $P(h\le x\le h))={\int }_h^{k}f(x)dx$

• The mode is the x value of the max point on the graph
• The median is the x value of the point where the area to it’s left on the PDF is $\frac{1}{2}$

Cumulative Distribution Functions:

• For every PDF (f(t)), there is a cumulative distribution function (CDF), defined as:

F(x)=P(X\leq x)=\int_{- \infty}^{\infty} f(t) , dt

$$-Itisa<u>function</u>thatgivesthearea\ast \ast totheleft\ast \ast ofanyxvalueonthe\ast \ast PDF\ast \ast .$$