In[]:=
fx[t_]=Piecewise[{{0,t<=0},{(8/3)*(6*t^3-6*t^2+1),t<1/2},{(8/3)*(-2*(t-1)^3),t<1},{0,t>=1}}]
Out[]=
0
t≤0
8
3
(1-6
2
t
+6
3
t
)
t<
1
2
-
16
3
3
(-1+t)
t<1
0
True
In[]:=
fy[t_]=Piecewise[{{0,t<=0},{(16/9)*Sqrt[3]-(16/3)*t+(64/27)*t^3,t<Sqrt[3]/2},{0,t>Sqrt[3]/2}}]
Out[]=
0
t≤0
16
3
3
-
16t
3
+
64
3
t
27
t<
3
2
0
True
In[]:=
xDist=ProbabilityDistribution[fx[t],{t,0,1}]
Out[]=
ProbabilityDistribution
0
x.≤0
8
3
1-6
2
x.
+6
3
x.

x.<
1
2
-
16
3
3
(-1+x.)
x.<1
0
True
,{x.,0,1}
In[]:=
yDist=ProbabilityDistribution[fy[t],{t,0,Sqrt[3]/2}]
Out[]=
ProbabilityDistribution
0
x.≤0
16
3
3
-
16x.
3
+
64
3
x.
27
x.<
3
2
0
True
,x.,0,
3
2

In[]:=
de=TransformedDistribution[Sqrt[(u-v)^2+(u-v)^2],{uxDist,vyDist}]
Out[]=
TransformedDistribution
2
2
(u-v)
,uProbabilityDistribution
0
x.≤0
8
3
1-6
2
x.
+6
3
x.

x.<
1
2
-
16
3
3
(-1+x.)
x.<1
0
True
,{x.,0,1},vProbabilityDistribution
0
x.≤0
16
3
3
-
16x.
3
+
64
3
x.
27
x.<
3
2
0
True
,x.,0,
3
2

In[]:=
dE[t_]=Evaluate[PDF[de,t],Assumptions->t<=1]
Out[]=
Sequence
-
9178
2
3
35
+
86612
2
567
t
1
2
(2
2
-
6
)
2
(-16781447+9688813
3
)
2835(-116+67
3
)
t
1
2
(-
2
+
6
)
10278833842771
2
-5934467009040
6
181440(-13801328+7968525
3
)
t
3
2
2
-
4(
2
(-4849+2013
3
)+1680
3
t+42
2
(-223+138
3
)
2
t
-1680
3
3
t
+210
2
(-7+5
3
)
4
t
+378
5
t
+84
2
6
t
-12
7
t
)
2835
0<t<
1
2
(2
2
-
6
)
8(4
2
(-59+105
3
)-28(-41+60
3
)t+84
2
(-13+15
3
)
2
t
-140(-7+6
3
)
3
t
+35
2
(-5+3
3
)
4
t
-21
5
t
+14
2
6
t
-2
7
t
)
2835
3
2
≤t<
2
1
2835(-116+67
3
)
2(-618717
2
+357291
6
+7(-655948+378689
3
)t+2898
2
(-750+433
3
)
2
t
+5810(-116+67
3
)
3
t
+6160
2
(-97+56
3
)
4
t
-840(-116+67
3
)
5
t
-112
2
(-116+67
3
)
6
t
+16(-116+67
3
)
7
t
)
1
2
(2
2
-
6
)<t<
1
2
(-
2
+
6
)
1
2835(-116+67
3
)
2(386407
2
-223001
6
+7(-231724+133745
3
)t-42
2
(-55738+32177
3
)
2
t
-1750(-116+67
3
)
3
t
-6160
2
(-97+56
3
)
4
t
-840(-116+67
3
)
5
t
+112
2
(-116+67
3
)
6
t
+16(-116+67
3
)
7
t
)
1
2
≤t<
3
2
(32(846205224
2
-488554344
6
+104(-34112844+19695239
3
)t+(4557657631
2
-2631352227
6
)
2
t
+(2817871917-1626949154
3
)
3
t
+6
2
(-926494367+534903394
3
)
4
t
+(768203686-443456953
3
)
5
t
+(1379767023
2
-796547009
6
)
6
t
+(-247856691+143045440
3
)
7
t
+10
2
(-9127696+5267087
3
)
8
t
+(4223250-2417169
3
)
9
t
+6
2
(-1665277+962012
3
)
10
t
-6(-71800+42103
3
)
11
t
-24
2
(-66107+38153
3
)
12
t
-24(-9222+5321
3
)
13
t
+48
2
(-657+379
3
)
14
t
+48(-116+67
3
)
15
t
))2835(-116+67
3
)
4
(-1+
3
+
2
t)
2
(4(9+
3
)-(
2
+7
6
)t+2
2
t
)

3
2
2
<t<
1
2
||
1
2
(-
2
+
6
)<t<
3
2
2
0
t≥
2
||t<0
Indeterminate
True
,Assumptionst≤1