Question

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

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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: