Spliced probability distribution modeling

Question

I have a simulation model for insurance claims that works as follows:

1. Assume random values come from two distributions defined as $$ \begin{eqnarray} f_1(x)&, & 0 < x \leq c \\ f_2(x)&, &x > c \end{eqnarray} $$
2. A random value will fall in $f_1(x)$ with probability $p$ and $f_2(x)$ with probability $1-p$.

So the model (in Excel) uses rand() to determine which distribution to sample from. $f_2(x)$ is shifted at $c$ so the minimum value from this distribution is $c$ while $f_1(x)$ is set to be capped at $c$. A determined number of values is generated in the model and then summed to give an aggregate value.

I'm trying express the limited expected value of this distribution and am wondering is the distribution simply defined as: $$ f_X(x) = \begin{cases} p \cdot f_1(x), & 0 < x \leq c \\ (1-p) \cdot f_2(x), & x >c \end{cases} $$ or am I missing something? This distribution will not be continuous at $c$ unless I force it to through solving for $p$ but assume that's not a requirement.

Answer 1

Hint: You can treat both functions as truncated distributions with $f_{1}(x)$ as truncated in the interval $0\le x\le c$ and the other $f_{2}(x)$ as another truncated in the interval $x\ge c$. With both $f_{1}(x)$ and $f_{2}(x)$ being probability distributions themselves, using linearity of expectation and p and 1-p as constants, you can, I believe, treat them such as this $E(x) = pE(x_1) + (1-p)E(x_2)$ where $E(x_1)$ and $E(x_2)$ are expected values of truncated functions $f_{1}(x)$ and $f_{2}(x)$ provided these functions are deliberately capped at these intervals while the original distributions run in the range $-\infty \le x\le \infty$.