3.2 The Continuous Case
Theorem 3.2.1 (Convolution formula). Let \(X\) and \(Y\) be independent continuous random variables with densities \(f_X\) and \(f_Y\). Then \(Z = X+Y\) has density \[f_Z(z) = \int _{-\infty }^{\infty } f_X(z-y)\,f_Y(y)\,dy .\]
Proof. Make the change of variables \[z = x+y,\qquad v = y,\qquad \text {so that}\qquad x = z-v,\quad y = v .\] The Jacobian of the inverse transformation is \[\left |\frac {\partial (x,y)}{\partial (z,v)}\right | = \left |\begin {matrix} 1 & -1\\ 0 & 1\end {matrix}\right | = 1 ,\] so the joint density of \((Z,V)\) is \[f_{Z,V}(z,v) = f_{X,Y}(z-v,\ v) = f_X(z-v)\,f_Y(v),\] the factorisation being independence. Integrating out the variable that is not wanted, \[f_Z(z) = \int _{-\infty }^{\infty } f_{Z,V}(z,v)\,dv = \int _{-\infty }^{\infty } f_X(z-v)\,f_Y(v)\,dv,\] which is the stated formula with \(v\) renamed \(y\). □
Note. The formula is symmetric in \(X\) and \(Y\) — substituting \(u=z-y\) turns it into \(\int f_X(u)f_Y(z-u)\,du\) — so either variable may be the one integrated out. Choose whichever has the simpler density, or the simpler range: in practice the limits of integration cause more difficulty than the integrand, because the requirement that both \(f_X(z-y)\) and \(f_Y(y)\) be non-zero restricts \(y\) to an interval that usually depends on \(z\).
Example 3.2.2. Let \(X\) and \(Y\) be independent, each uniform on \((0,1)\). Find the density of \(Z=X+Y\).
Solution. Both densities equal \(1\) on \((0,1)\) and vanish elsewhere, so the integrand is \(1\) exactly when \(0<y<1\) and \(0<z-y<1\), that is when \[\max (0,\ z-1) < y < \min (1,\ z),\] and \(z\) ranges over \((0,2)\). The density is therefore the length of that interval: \[f_Z(z) = \begin {cases} z, & 0<z\leq 1,\\ 2-z, & 1<z<2,\\ 0, & \text {otherwise.} \end {cases}\] This is the triangular density on \((0,2)\). It is worth noticing that the sum of two flat densities is not flat — values near \(1\) arise in many ways and values near \(0\) or \(2\) in few — which is the simplest instance of the concentrating effect that the central limit theorem describes in general.
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