Survivor function of a variable that has discrete and continuous components

The function $F(t) = \mathsf P(T>t) = 1-\mathsf P(T\leq t)$ is clearly of RCLL class on $[0,\infty)$. As a result, the definitions of continuous part of the hazard function $\lambda_c$ and discrete parts allow you computing $F$ by integrating $\lambda_c$ in between of the jumps, and applying jump conditions at $t = a_j$. The latter have the following shape: $$ \lambda_j = \mathsf P(T = a_j\mid T\geq a_j) = \frac{F(a_j-) - F(a_j)}{F(a_j-)}\implies F(a_j) = F(a_j-)(1-\lambda_j) $$ where $F(t-):=\lim_{s\uparrow t}F(s)$.

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Probability Theory

Probability Distributions

Random Variables

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