Ex 1: Distribution of distance of two cities if cities distributed uniformly in
plane.
p HdL = p
2
2
Ixi - xj M + Iyi - yj M
Two formulae we use.
1. If z =f(x,y), we can find the pdf of z, pz HzL by marginalizing the joint pdf p(x,y,z) using the
convolution formula: pz HzL = Ù dx dy pHx, y, zL = Ù dx Ù dy∆Hz - f Hx, yLL pX HxL pY HyL
2. Change of variables for densities: pz HzL dz = fA HaL da
Distribution of difference of two uniform variables in one dimension
In[1230]:=
Integrate@ DiracDelta@z - Hx - yL * 1 * 1D, 8x, 0, 1<, 8y, 0, 1<D
distr1 = Assuming@z > - 1 && z < 1, %  FullSimplifyD
Integrate@distr1, 8z, - 1, 1<DH* normalized *L
Plot@distr1, 8z, - 1, 1<D
Out[1230]=
- H1 + zL HeavisideTheta@- 1 - zD - H- 1 + zL HeavisideTheta@1 - zD + 2 z HeavisideTheta@- zD
Out[1231]=
1 - z + 2 z HeavisideTheta@- zD
Out[1232]=
1
1.0
0.8
0.6
Out[1233]=
0.4
0.2
-1.0
-0.5
0.5
1.0
Distribution of SQUARED difference of two uniform variables by change of variable. Due to ambiguity at z=0, change coordinates separately for z>0 and z<0. Pick up multiplicity of 2 in addition to Jacobian determinant
2
Session09.nb
1
In[1234]:=
distr2 = Jdistr1 . z ®
*2
2
distr2 = AssumingB
1
s N*
s
s > 0, Simplify@%DF
distr2Fct@x_D := If@0 £ x && x £ 1, distr2 . s ® x, 0D
Integrate@distr2, 8s, 0, 1<DH* normalized? *L
Plot@distr2, 8s, 0, 1<D
1-
s +2
s HeavisideThetaB-
s F
Out[1234]=
s
1
Out[1235]=
-1 +
s
Out[1237]=
1
3.0
2.5
2.0
Out[1238]= 1.5
1.0
0.5
0.2
0.4
0.6
0.8
1.0
Distribution of sum of squared differences again from convolution
In[1239]:=
distr3 = Integrate@distr2Fct@sD * distr2Fct@ Α - sD, 8s, 0, 1<D
-3 + Π
Π-4
Out[1239]=
-2 + 4
0
ΑŠ1
Α +Α
0<Α<1
- 1 + Α - Α + 2 ArcCscB Α F - 2 ArcTanB - 1 + Α F 1 < Α < 2
True
Session09.nb
In[1240]:=
Out[1240]=
Integrate@distr3, 8Α, 0, 2<D
Plot@distr3, 8Α, 0, 2<D
1
3.0
2.5
2.0
Out[1241]= 1.5
1.0
0.5
0.5
1.0
Transform to distance coordinates, d =
In[1242]:=
1.5
2.0
Α . Define separately for 0 < d < 1 and 1 < d <
distr4 = PiecewiseB:9Idistr3@@1, 2, 1DD . Α ® d2 M * 2 d, H0 < d && d £ 1L=,
:Idistr3@@1, 3, 1DD . Α ® d2 M * 2 d, J1 < d && d <
:0, d < 0 ÈÈ d >
2 d d2 - 4
Out[1242]=
2 N>,
2 > >F
d2 + Π
2 d - 2 - d2 + 4
0 < d && d £ 1
- 1 + d2 + 2 ArcCscB
d2 F - 2 ArcTanB
0
- 1 + d2 F
1 < d && d <
True
Final result
In[1243]:=
2
PlotBdistr4, :d, 0,
2 >, AxesLabel ® 8"d", "pHdL"<F
pHdL
1.4
1.2
1.0
0.8
Out[1243]=
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
1.2
Nice, distr. is differentiable in d=1 even though it is not obvious
1.4
d
2
3
4
Session09.nb
In[1275]:=
deriv = D@distr4, dD
Limit@ deriv@@1, 2, 1DD, d -> 1D
Limit@ deriv@@1, 4, 1DD, d -> 1D
0
d<0
2d 2d-
4d
+ 2 d2 - 4
d2 + Π
0<d<1
d2
2 H- 5 + ΠL
Out[1275]=
dŠ1
2
-2 d 2 d +
1-
1
d
d2
2 2 + d2 - 4
2
+
d
d2
4d
-
-1+d2
1<d<
-
- 1 + d2 - 2 ArcCscB
d2 F + 2 ArcTanB
- 1 + d2 F
0
Indeterminate
Out[1276]=
2 H- 5 + ΠL
Out[1277]=
2 H- 5 + ΠL
d Š 2 ÈÈ d >
True
Plot how the first expression becomes negative
In[1296]:=
distr4@@1, 2, 1DD
PlotB:2 d d2 - 4
d2 + Π ,
2 d - 2 - d2 + 4
Out[1296]=
2 d - 2 - d2 + 4
- 1 + d2 + 2 ArcCscB
d2 F - 2 ArcTanB
- 1 + d2 + 2 ArcCscB
d2 F - 2 ArcTanB
0.4
1.0
- 1 + d2 F >, :d, 0,
- 1 + d2 F
1.0
0.5
Out[1297]=
0.2
2
-1+d2
0.6
0.8
1.2
1.4
-0.5
-1.0
-1.5
distribution of sum of standard uniforms = distribution of difference of shifted uniforms
2 >F
2
Session09.nb
In[1256]:=
Integrate@ DiracDelta@z - Hx + yL * 1 * 1D, 8x, 0, 1<, 8y, 0, 1<D
distr1 = Assuming@z > 0 && z < 2, %  FullSimplifyD
Integrate@distr1, 8z, - 0, 2<DH* normalized *L
Plot@distr1, 8z, 0, 2<D
Out[1256]=
2 H- 1 + zL HeavisideTheta@1 - zD - H- 2 + zL HeavisideTheta@2 - zD - z HeavisideTheta@- zD
Out[1257]=
2 - z + 2 H- 1 + zL HeavisideTheta@1 - zD
Out[1258]=
1
1.0
0.8
0.6
Out[1259]=
0.4
0.2
0.5
1.0
1.5
2.0
5