Continuous distribution with PDF a k^a x^(a-1) on support (0 1/k]
PowerDistribution[k, a]
Mean[PowerDistribution[k, a]]
→ a/(k + a*k)PDF[PowerDistribution[k, a], x]
→ Piecewise[{{a*k^a*x^(-1 + a), Inequality[0, Less, x, LessEqual, k^(-1)]}}, 0]CDF[PowerDistribution[k, a], x]
→ Piecewise[{{(k*x)^a, Inequality[0, Less, x, LessEqual, k^(-1)]}, {1, x > k^(-1)}}, 0]