I have a probability density function: $f(x) = \frac{\beta}{2} e^{-\beta |x|}$ with $\beta \gt 0$ and I've been asked to find the cumulative distribution function F of X. So I know in order to get from a pdf to cdf that I need to integrate.
But when I integrate I'm not 100% sure what to do, does it just become: $F(x) = -\frac{\beta e^{-\beta |x|}}{2\beta}$
I wasn't sure whether I needed to consider it for x being positive and x being negative. But I didn't think it would affect the outcome, so I just integrated normally.
Finally I need to sketch the CDF F of X. & I have no idea how to do this. I think I know what the PDF would look like but I'm unsure on what the CDF would look like.
For the cumulative distribution function, you need to integrate the density from $-\infty$ (or the lowest point in the support) to the point of interest. So you want $$F(x)=\int\limits_{y=-\infty}^{x} \frac \beta 2 e^{-\beta |y|} \,dy$$
For $x \le 0$ you can say $$F(x)=\int\limits_{y=-\infty}^{x} \frac \beta 2 e^{\beta y}\, dy=\frac12 e^{\beta x} =\frac12 e^{-\beta |x|}$$ but for $x > 0$ you need to deal with the change in sign of $|y|$ to get $$F(x)=\int\limits_{y=-\infty}^{0} \frac \beta 2 e^{\beta y}\, dy + \int\limits_{y=0}^{x} \frac \beta 2 e^{-\beta y}\, dy =\frac12 +\left(-\frac12 e^{-\beta x} +\frac12\right)= 1 - \frac12 e^{-\beta |x|} $$