A continuous random variable has a pdf of the form
$f(x)=\begin{cases}ce^{-2x}&x>1\\0&x\leq1,\end{cases}$where is a constant. Find the value of and the cdf of .
If is a pdf, then we must have , so we simply need to solve for the value of . That is,
$\int_{-\infty}^\infty f(x)\,dx$ $=\int_{-\infty}^\infty ce^{-2x}\,dx$ $=\int_1^\infty ce^{-2x}\,dx$ $=\left.-\frac{c}{2}e^{-2x}\right|_{x=1}^\infty$ $=\frac{c}{2}e^{-2}=1\Rightarrow c=2e^2.$
Then the cdf of is
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Pdf and cdf of a continuous random variable
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