In[]:=
x=TransformedDistribution[u^(1/2)*Cos[v],{uUniformDistribution[{0,1}],vUniformDistribution[{0,2*Pi}]}]
Out[]=
TransformedDistribution
x.1
Cos[x.2],{x.1UniformDistribution[{0,1}],x.2UniformDistribution[{0,2π}]}
In[]:=
PDF[x,t]
Out[]=
2
1-
2
t
π
-1<t<1
0
True
In[]:=
y=TransformedDistribution[u^(1/2)*Sin[v],{uUniformDistribution[{0,1}],vUniformDistribution[{0,2*Pi}]}]
Out[]=
TransformedDistribution
x.1
Sin[x.2],{x.1UniformDistribution[{0,1}],x.2UniformDistribution[{0,2π}]}
In[]:=
PDF[y,t]
Out[]=
2
1-
2
t
π
-1<t<1
0
True
In[]:=
f[t_]=Piecewise[{{0,t<=-1},{(2/Pi)*(Sqrt[1-t^2]),-1<t<1},{0,t>=1}}]
Out[]=
0
t≤-1
2
1-
2
t
π
-1<t<1
0
True
In[]:=
Plot[f[t],{t,-1,1},AspectRatio->1/2]
Out[]=
In[]:=
Integrate[f[t],{t,-1,1}]
Out[]=
1
In[]:=
g=2*Integrate[f[tau]*f[t-tau],{tau,-Infinity,Infinity},Assumptions->{t>=0,t<=2}]
Out[]=
2
0
t≥2||t<0
Integrate
0
t-tau≤-1
2
1-
2
(t-tau)
π
-1<t-tau<1
0
True
0
tau≤-1
2
1-
2
tau
π
-1<tau<1
0
True
,{tau,-∞,∞},Assumptions0≤t<2
True