Probability

$\frac{\alpha^{n-1}}{\beta^n}(n \beta-n \alpha+\alpha)$

$\frac{(\beta-\alpha)^{n-1}}{\beta^n}(n \alpha+\beta-\alpha)$

$\frac{\alpha^n}{\beta^n}(n \alpha+\beta)$

$\left(\frac{\beta-\alpha}{\beta}\right)^n(n \beta+n \alpha+1)$