Histogram of signal with a time shift added
In your example, $x_3[n]$ is the sum of two iid random variables, $x_1[n]$ and $x_1[n-1]$. Since each is taken from the uniform distribution, the probability density function (pdf) of each is:

$$
f_{x_1}(x) = \begin{cases}
\frac{1}{b-a}, &amp; x \in [a,b] \\
0, &amp; \text{otherwise}
\end{cases}
$$

This is just a rectangle function that is scaled and shifted to lie on the interval $[a,b]$. One property of the sum of independent random variables is that the pdf of the sum is equal to the convolution of the individual random variables' pdfs. Therefore:

$$
f_{x_3}(x) = f_{x_1}(x) * f_{x_1}(x)
$$

If you convolve two rectangles together, you get a triangle function. Therefore, if you histogram enough samples of $x_3[n]$, you should see the plot converge to a triangular shape.