Cumulative Distribution Function of $X+Y$, where $X,Y$ are independent is convolution of $F_X$ and $F_Y$?

Question

I'm reading Introduction to Probability Models by Sheldon Ross, 12th edition. On page 57, it says:

Suppose first that $X$ and $Y$ are continuous, $X$ having probability density $f$ and $Y$ having probability density $g$. Then, letting $F_{X+Y}(a)$ be the cumulative distribution function of $X+Y$, we have: \begin{align*} F_{X+Y}(a) &= P \{ X + Y \le a \} \\ &= \iint_{x+y \le a} f(x) g(y) \, dx \, dy \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{a-y} f(x) g(y) \, dx \, dy \\ &= \int_{-\infty}^{\infty} \left( \int_{-\infty}^{a-y} f(x) \, dx \right) g(y) \, dy \\ &= \int_{-\infty}^{\infty} F_X(a-y) g(y) \, dy \\ \end{align*} The cumulative distribution function $F_{X+Y}$ is called the convolution of the distributions $F_X$ and $F_Y$ (the cumulative distribution functions of $X$ and $Y$, respectively).

I follow the equations, but that looks like the convolution of $F_X$ and $g$, not $F_X$ and $F_Y$. Standard, general convolution for general functions $f$ and $g$ is:

\begin{align*} (f * g)(t) &= \int_{-\infty}^\infty f(\tau) g(t - \tau) \, d \tau \\ \end{align*}

It seems like $F_{X+Y}(a) = F_X * g$ is the correct relationship rather than $F_{X+Y}(a) = F_X * F_Y$.

Can someone explain this? Does convolution have a more flexible definition? Or is the textbook mistaken?

Answer 1

As @Priyatham confirmed, the textbook simply made a mistake.

Answer 2

The book is a bit confusing but essentially correct, the author is mixing the terms cumulative distribution function with just distribution. Usually the distribution of a random variable is defined as the measure induced by this random variable, that is, if $P$ is the probability measure in $\Omega $ and $X:\Omega \to \mathbb{R}$ is a random variable, then $P_X:=P\circ X^{-1}$ is a probability measure in $\mathbb{R}$ induced by $X$.

It is usually this measure $P_X$ what is known as the distribution or law of $X$, and $F_X$ is known as the distribution function or the cumulative distribution function of $X$. Now, if $\mu$ and $\nu$ are two finite measures in a topological group $(G,\odot)$ then it convolution is defined by

$$ \mu*\nu(A):=\int_{A}\mu(A\odot x^{-1})\nu(\mathop{}\!d x) $$

where $x^{-1}$ is the inverse (respect to the group operation) of $x$ and $A\odot x^{-1}:=\{y\odot x^{-1}\in G:y\in A\}$. By example you have that

$$ F_{X+Y}(a)=P_X*P_Y((-\infty ,a]) $$

as $\mathbb{R}$ is a topological group under the operation of addition. In general

$$ P_X*P_Y(A)=\int_{A}P_X(A-x) P_Y(dx)=P_{X+Y}(A) $$

for any Borel set $A\subset \mathbb{R}$. The conclussion is that the author did the mistake of mix both concepts of cumulative distribution function and distribution.