The problem states that $X$ and $Y$ are independent random variables uniformly distributed on the interval $(-1, 1)$, and it asks to prove that $Z = X + Y$ is an absolutely continuous random variable with the given probability density function (PDF):

$f_Z(z) = \frac{1}{4}(2 - |z|) \cdot \mathbf{1}_{[-2, 2]}(z)$

Step-by-Step Approach:

1. Sum of Independent Uniform Random Variables: We are tasked with finding the PDF of the sum $Z = X + Y$, where $X$ and $Y$ are independent and uniformly distributed over $(-1, 1)$. The PDF of each random variable, $X$ and $Y$, is:

   $f_X(x) = f_Y(y) = \mathbf{1}_{[-1, 1]}(x)$ This indicates that the random variables $X$ and $Y$ are uniformly distributed, with their values ranging from $-1$ to $1$.

2. Convolution of PDFs: Since $Z = X + Y$ is the sum of two independent random variables, the PDF of $Z$ can be found using the convolution of the PDFs of $X$ and $Y$:

   $f_Z(z) = (f_X * f_Y)(z) = \int_{-\infty}^{\infty} f_X(x) f_Y(z - x) dx$

   Given that $f_X(x) = \mathbf{1}_{[-1, 1]}(x)$ and $f_Y(y) = \mathbf{1}_{[-1, 1]}(y)$, the convolution integral simplifies to:

   $f_Z(z) = \int_{-1}^{1} \mathbf{1}_{[-1, 1]}(z - x) dx$

   This integral calculates the area of overlap between the intervals where the indicator functions are non-zero.

3. Support of $Z$: The sum of two random variables $X$ and $Y$ uniformly distributed on $(-1, 1)$ will have a support $Z \in [-2, 2]$. Therefore, the PDF of $Z$ will be non-zero only on this interval.

4. Computing the PDF: The convolution results in the following piecewise function for $f_Z(z)$:

   \begin{cases} \frac{1}{4}(2 - |z|) & \text{for } z \in [-2, 2] \\ 0 & \text{otherwise} \end{cases}$$ This matches the given PDF $$f_Z(z) = \frac{1}{4}(2 - |z|) \cdot \mathbf{1}_{[-2, 2]}(z)$$.

Thus, $Z = X + Y$ is an absolutely continuous random variable with the specified PDF.

Would you like more details on any step, or have questions on the derivation?

Here are some related questions for further exploration:

1. How does the convolution of two independent random variables' PDFs work in general?
2. How would the solution change if $X$ and $Y$ had different distributions (not uniform)?
3. Can we compute the cumulative distribution function (CDF) for $Z$?
4. How would you simulate the distribution of $Z$ using a programming language like Python?
5. What is the interpretation of the piecewise function in the context of probability distributions?

Tip: When performing convolution of PDFs, always check the overlap between the ranges of the two random variables. This will help in setting the limits of integration.