Integrating product of three pdf (triangular dist. is disturbing)
- To: mathgroup at smc.vnet.net
- Subject: [mg79321] Integrating product of three pdf (triangular dist. is disturbing)
- From: negedea at googlemail.com
- Date: Tue, 24 Jul 2007 06:11:25 -0400 (EDT)
Hello All,
I have a difficulty in integrating the product of three probability
density functions f1, f2 and f3. From negative infinity to infinity
over the variables forming f2 and f3, i.e with respect to dy and dz.
The problem seems to arise because f1 is triangular distribution. I
got the code for triangular distribution from this this user group.
The code used for the triangular distribution pdf diginated by f1 is:
In[26]:=
trianglePDF[min_, max_, mode_, x_] :=
Piecewise[{{(2*(-min + x))/((max - min)*(-min + mode)),
min <= x <= mode}, {(2*(max - x))/((max - min)*(max -
mode)),
mode < x <= max}}]
In[27]:=
f1=trianglePDF[871.87, 1362.3, 1285.2, x] //InputForm
As a result f1 will take the form:
Out[27]//InputForm=
Piecewise[{{9.866339130123132*^-6*(-871.87 + x),
871.87 <= x <= 1285.2},
{0.00005289304737553565*(1362.3 - x),
Inequality[1285.2, Less, x, LessEqual, 1362.3]}},
0]
Actually X in f1 stands for the expression:
{{x -> -0.3052676799266343*Cot[0.017453292519943295*y]*
(1.*=CF=841 - 724.0368939942928*Tan[0.017453292519943295*y] -
55.69524532558123*z*Tan[0.017453292519943295*y])}}
when this replacement is done f1 will take the form:
Piecewise[{{9.866339130123132*^-6*(-871.87 + x),
871.87 <= x <= 1285.2},
{0.00005289304737553565*(1362.3 - x),
Inequality[1285.2, Less, x, LessEqual, 1362.3]}},
0]/.x -> -0.3052676799266343*Cot[0.017453292519943295*y]*
(1.*=CF=841 - 724.0368939942928*Tan[0.017453292519943295*y] -
55.69524532558123*z*Tan[0.017453292519943295*y])
f2 is a normaly distributed variable y with mean =42.13 and std = 3.
i=2Ee,
In[22]:=
<<Statistics`NormalDistribution`
f2dist=NormalDistribution[42.13,3];
f2=PDF[f2dist,y]//InputForm
As a result f2 will take the form (written in InputForm):
Out[24]//InputForm=
1/(3*E^((-42.13 + y)^2/18)*Sqrt[2*Pi])
f3 is a pdf equation for a weibule distributed variable z with
parameters alpha = 13.13 and beta = 29.8 and a location parameter to
change it to standard distribution L.
In[6]:=
<<Statistics`ContinuousDistributions`
f3dist=WeibullDistribution[13.13, 29.8]
In[20]:=
L=-11.12;
f3=PDF[f3dist,z-L]//InputForm
As a result f3 will take the form (written in InputForm):
Out[21]//InputForm=
(5.778367760333042*^-19*(11.12 + z)^12.13)/
E^(4.4008893833458054*^-20*(11.12 + z)^13.13)
What I wanted is to integrate the product f1*f2*f3 with respect to dy
and dz over the interval -infinity to infinity.
Please, do help me.
Thaning you in advance!
Best regards,
Negede