Convolution for sums of Unif[0,1] random variables

Let (Subscript[f, X]) and (Subscript[f, Y]) be pdfs of Unif[0,1] random variables. The red interval is the graph of (Subscript[f, X])(x). The green interval is the graph of (Subscript[f, Y])(w-x). The blue graph is the graph of the product (Subscript[f, X])(x)(Subscript[f, Y])(w-x).

f[x_]:=Piecewise[{{1,0≤ x≤ 1}}];
Manipulate[Show[
Plot[1,{x,0,1},PlotStyle->Directive[Red,Opacity[0.5],Thickness[0.02],CapForm[None]]],
Plot[1,{x,w-1,w},PlotStyle->Directive[Green,Opacity[0.5],Thickness[0.02],CapForm[None]]],
Plot[f[x]×f[w-x],{x,-1.5,2.5},Filling->Axis,PlotStyle->Directive[Blue],PlotPoints->400],
PlotRange->{{-1.5,2.5},{-0.1,1.2}},AxesLabel->{"x","(Subscript[f, X])(x), (Subscript[f, y])(w-x)"}],
{w,0,2}]

f[x_]:=Piecewise[{{1,0≤ x≤ 1}}];
Manipulate[GraphicsRow[{
Show[
Plot[1,{x,0,1},PlotStyle->Directive[Red,Opacity[0.5],Thickness[0.02],CapForm[None]]],
Plot[1,{x,w-1,w},PlotStyle->Directive[Green,Opacity[0.5],Thickness[0.02],CapForm[None]]],
Plot[f[x]×f[w-x],{x,-1.5,2.5},Filling->Axis,PlotStyle->Directive[Blue],PlotPoints->400],
PlotRange->{{-1.5,2.5},{-0.1,1.2}},AxesLabel->{"x","(Subscript[f, X])(x), (Subscript[f, y])(w-x)"}],
Plot[Piecewise[{{t,0≤t≤1},{2-t,1<t≤2}}],{t,-1,w},PlotRange->{{-1,2.5},{-0.1,1.2}},AxesLabel->{"w","(Subscript[f, W])(w)"}]
}],{w,0,2.5}]

Mathematica can also compute the convolution of two functions

Here is the pdf of a Unif[0,1] distribution:

f[x_]=PDF[UniformDistribution[],x]

The convolution of f with itself produces the pdf of the sum of two Unif[0,1] random variables:

g[x_]=Convolve[f[t],f[t],t,x]

Plot[g[x],{x,-1,3}]

The convolution of g with f produces the pdf of the sum of three Unif[0,1] random variables:

h[x]=Convolve[g[t],f[t],t,x]

Plot[h[x],{x,-1,4}]