In[2]:= genExpDist =
ProbabilityDistribution[(1 - E^(-\[Lambda] x))^\[Alpha], {x, 0,
Infinity}]
Out[2]= ProbabilityDistribution[(1 -
E^(-\[FormalX] \[Lambda]))^\[Alpha], {\[FormalX], 0, \[Infinity]}]
In[7]:= \[Lambda] = 1; \[Alpha] = 2;
In[8]:= RandomVariate[genExpDist]
Out[8]= 2.01697