Calculations about Pareto distribution

Question

We assume that the random variable $Y$ is a Pareto distribution with parameter $y_o$ and $ a\gt 0$: $$P(Y\leq y)= \left(1 - \left(\frac{y_o}{y}\right)^a\right)\cdot\mathsf 1_{[y_0,\infty)}(y).$$ What is the distribution of $Y$ conditioned on $\{Y>M\}$ (where $M>y_0$)?

Answer 1

By the definition of conditional probability, for $y>M$ we have \begin{align} \mathbb P(Y\leqslant y\mid Y>M) &= \frac{\mathbb P(Y\leqslant y, Y>M)}{\mathbb P(Y>M)}\\ &= \frac{\left(1-\left(\frac{y_0}y \right)^a\right) - \left(\frac{y_0}M \right)^a}{\left(\frac{y_0}M \right)^a}. \end{align}