Now remember that U is in (0,1). So ln(U)<0. Hence -ln(U)/theta>0
So if t is 0, the probability is 0. So let's see for t>0
Now, remember that the exponential function is strictly increasing over the real numbers.
Now remember that t>0. So , since theta*t>0
And , since an exponential is always positive.
Hence, by definition of the cumulative distribution function of a uniformly distributed rv,
And finally, we have :
, if t>0.
If t<0, then the probability is 0.
This is the exact cdf of an exponential distribution !
It's referred as the "inversion formula" in my notes, but I don't know how you guys call it.