- #1
TeenieBopper
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Homework Statement
Find the CDF of [tex]f(x) =
|\frac{x}{4}| if -2<x<2 \\
0 otherwise
[/tex]
Homework Equations
The Attempt at a Solution
I have to integrate the pdf and to do so, I have to split it into two parts
[tex]\int_{-x}^{0}\frac{-t}{4}dt + \int_{0}{x}\frac{t}{4}dt[/tex]
integrating I get [tex]\frac{x^2}{8} + \frac{x^2}{8} = \frac{x^2}{4}[/tex]
This isn't a strictly increasing function, which is a requirement to be a CDF. So I need to break it into cases. This is where I'm running into trouble (sorry, I don't know if/how to use cases environment here).
F(x)
[tex]
0 if x < -2 \\
1-\frac{x^2}{4} if -2<x<0 \\
some equation if 0<x<2 \\
1 if x>2
[/tex]
I know that the CDF must have a value of .5 if X=0, but I'm not sure how to set up the rest of the cases so that F(0)=.5 and F(2)=1
Am I allowed to keep the two integrals from above "split"? Because if I am, then the following should work:
F(x)=
[tex]
0 x<-2 \\
\frac{1}{2} -\frac{x^2}{8} if -2<x<0 \\
\frac{1}{2} + \frac{x^2}{8} if 0<x<2 \\
1 if x > 2
[/tex]
Am I just overthinking it, or can I just use that for the CDF?
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