A probability problem generated by http://www.saab.org/actuarial.cgi

Question

anonymous · Answer

attachment

anonymous · Answer

Would you need to integrate x f(x) from 0 to X ?

anonymous · Answer

You need to find the pdf of the distribution first and see what kind of random variable we are dealing with

anonymous · Answer

Here is the solution generated by http://saab.org/actuarial.cgi

anonymous · Answer

So we have $X\sim\text{Exp}(\beta=7)$, so 
$$f_X(x)=\begin{cases}\frac{1}{7}e^{-x/7}&\text{for }x>0\\0&\text{else}\end{cases}$$
I'll try it out with the method of distribution functions. If $f_X(x)$ is the pdf, then the cdf is
$$\begin{align*}F_X(x)&=\int_{-\infty}^xf_X(t)~dt\
&=\frac{1}{7}\int_0^xe^{-t/7}~dt\
&=1-e^{-x/7}
\end{align*}$$
Now to find the cdf of $U$:
$$\begin{align*}F_U(u)&=P(U\le u)\
&=P([X]\le u)
\end{align*}$$
Here's a plot of the random variables:|dw:1391573504942:dw|