Suppose $X$ and $Y$ are independent uniformly distributed on the interval $[-a/2,a/2]$. What is the density function of $Z=X+Y$ and of $Z=X-Y$?
I know that it will be the convolution of densities $f_X$ and $f_Y$. I think $f_X$ is $1/a$ when $x\in [-a/2, a/2] $. But I cannot determine the limits for integration of $f_Y$.
Best Answer
Hint: We have $f_X(x) = f_Y(y) = 1_{(-a/2,a/2)}$ that is 1 on $(-a/2,a/2)$ and 0 elsewhere.
From Convolution Theorem and Marginal Density Intuition.:
So determine
$$f_Z(z) = \int_{-\infty}^{\infty} f_Y(z \pm x) f_X(x) dx$$
$$f_Z(z) = \int_{-\infty}^{\infty} f_Y(z \pm x) \frac{1}{a} 1_{(-a/2,a/2)}dx$$
$$ = \int_{-a/2}^{a/2} f_Y(z \pm x) (\frac{1}{a})dx$$
So what is $f_Y(z \pm x)$ ?
See more: Convolution of probability distributions