Let $X, Y$ be independent random variables, both uniformly distributed over the interval $(0,1)$. That is, $$f_{X}(a)=f_{Y}(a) = \begin{cases} 1 & \text{if $0 < a <
1$}
\\ 0 & \text{otherwise} \end{cases}$$
Find $f_{X+Y}(a)$.
In my probability course, the solution to this problem was given as follows, but figuring out the integrals was very much "by inspection", and for this one, particular problem.
$$
f_{X+Y}(a) =
\begin{cases}
\int_{0}^{a} dy = a & \text{if $0 < a < 1$}
\\\int_{a-1}^{1} dy = 2 – a & \text{if $1 < a < 2$}
\end{cases}$$
It's not so apparent to me how we know these are the correct integration bounds in these two cases. I want a general, straightforward way to figure out exactly which integrals need to be computed for probability problems of this kind. So I read this answer by André Nicolas, which it finally made it clear to me how to start finding $P(X+Y \le a)$:
Another way: (Sketch) We can go after the cdf $F_Z(z)$ of $Z$, and then differentiate. So we need to find $\Pr(Z\le z)$.
For fixed $z$, draw the line with equation $x+y=z$. Draw the
square $S$ with corners $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$.Then $\Pr(Z\le z)$ is the area of the part $S$ that is "below" the
line $x+y=z$. That area can be calculated using basic geometry. There
is a switch in basic shape at $z=1$.
Then by drawing a figure with a square with vertices $(0,0), (0, 1), (1,0), (1,1)$, I come up with the following two integrals:
$$
\small
P(X+Y \le a) = \int_{0}^{1} \left( \int_{0}^{a-y} f(x) \ \mathrm{d} x \right) f(y) \ \mathrm{d}y = \int_{0}^{1} \left( a-y \right )1 \ \mathrm{d}y = a – \frac{1}{2} \ \ \ \ \ \text{if $0 < a < 1$}
$$
$$
\small
P(X+Y \le a) = \int_{0}^{1} \left( \int_{1-y}^{a-y} f(x) \ \mathrm{d} x \right) f(y) \ \mathrm{d}y = \int_{0}^{1} \left( (a-y) – (1 – y) \right )1 \ \mathrm{d}y = \int_{0}^{1} (a – 1) \ \mathrm{d}y \ \ \ \text{if $1 < a < 2$}
$$
It appears that I have made a mistake somewhere, since $\frac{d}{\mathrm{d}\ a}\left(F_{X+Y}(a) \right) =\frac{d}{\mathrm{d}\ a}\left( P(X+Y\le a) \right) = f_{X+Y}(a)$ in neither case gives me the correct answer for $f_{X+Y}(a)$.
Please help me see my mistake; what am I missing?
Best Answer
This problem is best solved by drawing diagram by plotting the line $x+y=z$ in 'x-y' plane and measuring the area under the line over the rectangle enclosed by $(0,0), (0,1), (1,0)$ and $(1,1)$ [covering $(x+y\leqslant z)$]. This is my recommendation. Alternatively, you may want to use method of convolution while calculating cdf of the sum X+Y exactly the way you have attempted (for details see this).
Looking at your approach I can see the issue in the range of y used in the integral when you try to find $P(X+Y\le a)$ for $0<a<1$. Unfortunately you can't write: $\int_{0}^{1} \left( \int_{0}^{a-y} f(x) \ \mathrm{d} x \right) f(y) \ \mathrm{d}y = \int_{0}^{1} \left( a-y \right )1 \ \mathrm{d}y$, since, when $0<X<a-y$, $0<a-Y<1$; thus, $a-1<Y<a$; but since we are evaluating the cdf for $0\le a\le1$, the range of Y eventually reduces to $0<Y<a$.
Similar sort of issue appears for the case $1\le a\le2$. Basically we just need to be a bit careful about the ranges so that the bounds used are valid ones.