Question

Verify $f\left(x\right)=\{\begin{array}{c}30x^4{e}^{-6x^5};x>0\\ 0;otherwise\end{array}$ for p.d.f if $f\left(x\right)$ is a p.d.f then find $P\left(1\right)$.

Answer 1

$p.d.f\text{of}f\left(x\right)={\int }_0^{\infty }30x^4{e}^{-6x^5}dx$

Let $6x^5=t\Rightarrow 30x^4=dt$, we get

$\Rightarrow {\int }_0^{\infty }{e}^{-t}dt=[-e^{-t}{]}_0^{\infty }=1$