# Integrating PDF of Continuous Uniform RV to get CDF

For a continous uniform RV $$\text{PDF} = \frac{1}{b-a} \text{ for } x\in[a,b)$$

(sorry cant figure out how to add whitespace)

and $$\text{CDF} = \int_{-\infty}^x \text{PDF} \, dx$$ so for $x>0$ I believe CDF of uniform RV should be $$\int_0^x \text{PDF} \, dx = \int_0^x \frac{1}{b-a} \, dx = \frac{x}{b-a}, \text{ from } x=x \text{ to } x=0$$ which equals just $$\frac{x}{b-a} \text{??}$$ But wikipedia says it should equal $$\frac{x-a}{b-a}$$ So where did I mess up with the integration?

• you can use \mbox{} to get regular text back in the middle of an equation. – Tony S.F. Apr 10 '16 at 1:23
• thanks, couldn't figure out how to google that – Austin Apr 10 '16 at 1:25

You should be integrating over the support of the PDF. The support is the set of all numbers such that the PDF is not zero. Therefore, your integral should be $\displaystyle\int_a^x \frac{1}{b-a} \, dx$. This will give CDF$=\dfrac{x}{b-a}-\dfrac{a}{b-a}=\dfrac{x-a}{b-a}$ just like wikipedia has.

The reason we integrate over the support is simple. We'd really like to integrate from $-\infty$ to $\infty$ but if the pdf is zero for all values outside of $[a,b]$ like in the case of the uniform distribution, we can reduce $\int_{-\infty}^\infty$ to just $\int_a^b$.

• Ahh, figured it would be a simple mistake that I was mentally blocking on. Thank you! – Austin Apr 10 '16 at 1:27
• Easily fixed, no problem. – Tony S.F. Apr 10 '16 at 1:29