 Solution

Let $r=1-p$. Sinceintegration from zero to $x>0$ gives $F(x)=(C/\beta )[\log (1-r{e}^{-\beta x})-\log p]$. Letting $x\to \infty$ and imposing unit total probability givesThe probability density function is positive for $x>0$ because $0<(1-p)e^{-\beta x}<1$. Consequently the cumulative distribution function is increasing, starts at zero and tends to one. This verifies that the derived normalizing constant defines a probability distribution. The family is the exponential-logarithmic distribution with the complementary mixing parameter $r=1-p$.