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

Question

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

$f(x)={\begin{cases}kx^{a-1}(1-x)^{b-1}&{\text{if }}0\leq x\leq 1,\\0&{\text{elsewhere}}.\end{cases}}$

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

Solution

${\begin{aligned}F(x)&=\int _{-\infty }^{x}f(t)dt\\&=\int _{0}^{x}12t(1-t)^{2}dt\\&=12\int _{0}^{x}(t-2t^{2}+t^{3})dt\\&=12{\Big (}{\frac {1}{2}}t^{2}-{\frac {2}{3}}t^{3}+{\frac {1}{4}}t^{4}{\Big )}{\Big |}_{0}^{x}\\&=x^{2}(6-8x+3x^{2})\end{aligned}}$

Discussion

Getting Started

Recall the first property of probability distribution functions:

$F(b)=\mathrm {Pr} (X\leq b)=\int _{-\infty }^{b}f(x)dx$

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