I have a set of i.i.d random variables $X_i$ with distribution: \begin{align} X\stackrel{d}{=} \begin{cases} \mathcal U[-z,z] & \text{ if } K_i \leq 0.5 \\ \mathcal U[-1,1] & \text{ if } K_i > 0.5 \end{cases} \end{align} where $K_i$ are i.i.d. with distribution $\mathcal U[0,1]$ and $z>0$.
How can I combine this into one smooth distribution $f(X_i)$?
The overall aim is I want to find the pdf of $X_i^2$, so I can use Chernoff to find the probability bounds $\Pr[\sum^N_{i=0} X^2_i \geq t] \leq 3e^{\frac{-t^2}{12\mu}}$
I was thinking if it is possible to rewrite it as something like $X_i \stackrel{d}{=} \mathcal U[0,z] ; $ and then additionally $X_i\stackrel{d}{=}X_i+\mathcal U[0,1-z] ; \text{ if } K<0.333$ and then possibly using some sort of convolution.
The density of $X_i$ is not smooth, but it is relatively simple
If $0 \lt z \le 1$ then $$f(x) = \left\{ \begin{array}{ll} \frac{1}{4z}+\frac14 & \mbox{if $-z \le x \le z$};\\ \frac14 & \mbox{if $-1 \le x \lt -z$ or $z \lt x \le 1$}\\ 0 & \mbox{if $x \lt -1$ or $1 \lt x$}\end{array} \right.$$
If $1 \le z$ then $$f(x) = \left\{ \begin{array}{ll} \frac{1}{4z}+\frac14 & \mbox{if $-1 \le x \le 1$};\\ \frac1{4z} & \mbox{if $-z \le x \lt -1$ or $1 \lt x \le z$}\\ 0 & \mbox{if $x \lt -z$ or $z \lt x$}\end{array} \right.$$
and in either case $\mathbb E\left[X\right] = 0$ and $\mathbb E\left[X^2\right] = \frac{z^2+1}6$