# Combined probability function

I have two PDFs given by:

fS1[x_?NumericQ] := PDF[NormalDistribution[1, 0.05], x];
FS1[x_?NumericQ] := CDF[NormalDistribution[1, 0.05], x];

fS2[x_?NumericQ] :=
PDF[GumbelDistribution[1 - EulerGamma*0.25*6^0.5/Pi, 0.25*6^0.5/Pi],
x];
FS2[x_?NumericQ] :=
CDF[GumbelDistribution[1 - EulerGamma*0.25*6^0.5/Pi, 0.25*6^0.5/Pi],
x];


And I want to compute the combined CDF:

FS12[x_?NumericQ] := NIntegrate[fS1[y]*FS2[x - y], {y, -Infinity, +Infinity}]
NIntegrate[FS12[x], {x, -Infinity, +Infinity}]


But the output I get is:

 NIntegrate::inumr: The integrand FS[x] has evaluated to non-numerical values for all sampling points in the region with boundaries {{0.,3.67005*10^28}}.


So I'm still struggling with FS12. Any additional ideas?

• You'll need to change all of the instances of [x]= to [x_]:= and Integrate to NIntegrate.
– JimB
Mar 2, 2020 at 4:55
• FS12 appears to be a CDF rather than a PDF.
– JimB
Mar 2, 2020 at 5:31
• @JimB Thank you! I also used ?NumericQ but I still get an error... Mar 2, 2020 at 5:46
• Quick experiment. Instead of your last NIntegrate do this Table[Print[{z,FS12[z]}],{z,-100,-50,5}]; and see what you get. Then think about why you would get that.
– Bill
Mar 2, 2020 at 6:07
• @Bill you are correct! When I constrained the ranges of integration the problem was solved! Thank you!! Mar 2, 2020 at 18:51