When we analyze data, we can observe several variables that may contain mutual information. For an example, there can be a binary variable such as $Y=\text{Have you ever smoked?}$. Then there will be a follow up question such as (in this case it is a continuous variable) "How old were you when you first smoked?"
For the variable X that measures "How old were you when you first smoked?",
\begin{align} X_1 &= \begin{cases} 0 & \text{if never smoked} \\ 1 & \text{if smoked} \end{cases} \\ X_2 &= \begin{cases} 0 & \text{if } x_1 = 0 \geq 0 & \text{if } x_1 = 1 \end{cases} \end{align}
So the distribution of $X_2$ will be like this:
That means it contains several zeros, as it is depends on the previous question ($X_1$).
One way to deal with this type of problem is calculate the age of first smoke ($X_2$) only for users (i.e., eliminating zeros). Then the drawback is that it will reduce the sample size with respect to $X_2$ variable.
Another way to model $X_2$ is convert it to a categorical variable. For an example someone can do like this:
\begin{equation} X_2^{\text{categorized}} = \begin{cases} \text{Never Smoke} & X_2 = 0 \\ \text{Young} & 0 < X_2 \leq 15 \\ \text{Middle} & 15< X_2 \leq 20 \\ \text{Old} & 20 < X_2 \end{cases} \end{equation}
But is there way to model $X$ by preserving the continuous nature using a mixture distribution? I mean mixture distribution in the sense that, this may be something like the product of $X_2$ and $X_1$. However, I am not sure how to do this.
Since in this case $X_1$ is binary, taking the product of $X_2$ and $X_1$ seems to make sense. But I am not sure how this will work in general, i.e., when $X_1$ has more than 2 categories.
