Science:MATH105 Probability/Lesson 2 CRV/2.08 CDF Example

Question

Consider the beta distribution function for X given in the last example,

$f(x)=\{\begin{array}{ll}k{x}^{a-1}(1-x{)}^{b-1}& \text{if }0\le x\le 1,\\ 0& \text{elsewhere}.\end{array}$

If a = 3 and b = 2, find the associated cumulative distribution function F(x) for x ≤ 1.

Solution

$\begin{array}{rl}F(x)& =\underset{-\mathrm{\infty }}{\overset{x}{\int }}f(t)dt\\ & =\underset{0}{\overset{x}{\int }}12t(1-t{)}^{2}dt\\ & =12\underset{0}{\overset{x}{\int }}(t-2t^2+t^3)dt\\ & =12(\frac{1}{2}{t}^2-\frac{2}{3}{t}^3+\frac{1}{4}{t}^4){|}_0^x\\ & =x^2(6-8x+3x^2)\end{array}$

Discussion

Getting Started

Recall the first property of probability distribution functions:

$F(b)=\mathrm{P}\mathrm{r}(X\le b)=\underset{-\mathrm{\infty }}{\overset{b}{\int }}f(x)dx$

We are given f(x) and need to find F, which can be found using the above integral.