> [!tldr] Zero-Inflated Distribution
> A **zero-inflated distribution** is the mixture between a point mass at $0$ and another (more regular) distribution. For example, a mixture with $\mathrm{Gamma}(3, 0.5)$ looks like
>
> ![[ZeroInflatedDistributionExample.png#invert|w60|center]]
> Here the point-mass has a weight of $p=1 / 6$.
>
>---
>
> This kind of distribution is common in real life, where there are two groups in a population: for example, if the variable is the number of kilometers driven in a car,
> - One group always have a response of $0$, e.g. children under 16;
> - Another group has a continuous distribution, e.g. the adults.
This kind of distribution can be modeled using [[Hurdle Models]].
```R fold title:"R code for generating the plot"
observations = c(rgamma(1000, shape = 3, rate = 0.5),
rep(0, 100))
data = data.frame(observations)
x = seq(0, 25, by = 0.01)
gamma_density = data.frame(x = x,
y = dgamma(x, 3, 0.5) * 1000)
ggplot() +
geom_histogram(data = data,
aes(observations,
fill = "Zero-Inflated Data"),
binwidth = 1) +
geom_line(data = gamma_density,
aes(x = x, y = y,
color = "Gamma Distribution")) +
scale_color_manual(name = "",
values = c("Gamma Distribution" = "#444444")) +
scale_fill_manual(name = "",
values = c("Zero-Inflated Data" = "#AAAAAA"))+
labs(x = "Value", y = "Count") +
my_theme
```