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Main questions are in bold but feel free to correct me if I'm wrong somewhere else. As far as possible, I need both intuition and formal explanation.

Let $X \sim Uniform(a,b)$ and $Y \sim Uniform(c,d)$ be two independent random variables and $Z=X-Y$ be the difference of these variables.

I need to compute $P(\mathbf{Z} \le z) = F_\mathbf{Z}(z)$ which I suppose is the PDF of a symmetric triangular distribution with support $[a-d,b-c]$. (a) Is this correct?

We know:

$$ f_\mathbf{X}(x) = \frac{1}{b-a} \ \ \ \ \ \ \ \ x \in [a,b] \\\\ f_\mathbf{Y}(y) = \frac{1}{d-c} \ \ \ \ \ \ \ \ y \in [c,d] $$

Let $Y' = -Y$, with:

$$ f_\mathbf{Y'}(y') = f_\mathbf{Y}(-y') = \frac{1}{d-c} \ \ \ \ \ \ \ \ y' \in [-d,-c] $$

(b) Is the above equation correct? Why?

We now have $Z=X+Y'$ and we know that the convolution/sum of random variables is as follows:

$$ f_\mathbf{Z}(z) = \int_{-\infty}^\infty f_\mathbf{X}(x)\ f_{Y'}(z-x)\ \ dx $$

I understand the convolution is the evolution in the common area below two functions where one is slipped on the x-axis (example). However, I don't understand (c) why it is equivalent to the addition of random variables.

I know how to draw the convolution graphically (example), from which I can deduce the analytic form. But (d) how can I compute $f_\mathbf{Z}(z)$ analytically?

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  • $\begingroup$ Why bother defining and obtaining density for $-Y$? To obtain CoV densities, you typically use Jacobian method or CDF method. Jacobian is a good choice. You are correct we can use heuristic arguments to establish the range/continuity of the densities. If your change of variable function is $f(x,y) = x-y$ calculate the gradient, and place the multivariate densities in vector form, etc. etc. to obtain the answer. So a, b look correct but c does not look correct. $\endgroup$
    – AdamO
    Commented Feb 29 at 18:18
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    $\begingroup$ I've taken liberty to add the self-study tag since I'd wager dollars-to-doughnuts this is a homework question. It will also garner a bit more patience from answerers who agree to provide didactic answers, rather than their own work. $\endgroup$
    – AdamO
    Commented Feb 29 at 18:21
  • $\begingroup$ Four different methods to compute this difference are described in detail at stats.stackexchange.com/questions/41467, thereby answering (a), (b), and (d). A specific example of this question is answered at stats.stackexchange.com/questions/545409, again answering (a), (b), and (d). (b) is answered generally at stats.stackexchange.com/questions/192807. Finally, the duplicate answers (c). $\endgroup$
    – whuber ♦
    Commented Feb 29 at 18:30
  • $\begingroup$ @AdamO That's not a homework, but what I really need is intuition, so the tag seems appropriate :) $\endgroup$
    – White1Hun
    Commented Feb 29 at 19:05
  • $\begingroup$ @whuber I haven't seen your last two links before and they helped me. I just will add this post to further complete the answer with an example. $\endgroup$
    – White1Hun
    Commented Feb 29 at 19:05

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