> [!tldr] Hurdle Models > A **hurdle model** of a variable $Y$ is a continuous distribution with an atom placed at $\{ Y=0 \}$. It is common for modeling data that follow a [[Zero-Inflated Distribution]]: for example, ![[ZeroInflatedDistributionExample.png#invert|w60|center]] > In this case, the hurdle model of the distribution of $Y$ is $\begin{align*} \mathbb{P}[Y=0]&= p,\\ \mathbb{P}[Y \le y ~|~ Y \ne 0]&= F_{Y'}(y) \end{align*}$for some CDF $F_{Y'}$. Equivalently, $Y=0\cdot Z+(1-Z)Y',$where $Z\sim \mathrm{Bernoulli}(p)$ for some $p \in (0,1)$, and $Y'\sim F_{Y'}$ controls the distribution of $Y$ when $Z=0$. In the example above, $p=1/6$ and $Y' \sim \mathrm{Gamma}(3, 0.5)$.