In[1]:= $Version
Out[1]= "10.2.0 for Microsoft Windows (64-bit) (July 28, 2015)"
In[2]:= Assuming[
Element[a, Reals] && Element[y, Reals] && Element[z, Reals],
Integrate[
DiracDelta[y - (x + a)/z] PDF[NormalDistribution[0, s],
x], {x, -Infinity, Infinity}]]
Out[2]= (E^(-((a - y z)^2/(2 s^2))) Abs[z])/(Sqrt[2 \[Pi]] s)
In[1]:= $Version
Out[1]= "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)"
In[2]:= Assuming[
Element[a, Reals] && Element[y, Reals] && Element[z, Reals],
Integrate[
DiracDelta[y - (x + a)/z] PDF[NormalDistribution[0, s],
x], {x, -Infinity, Infinity}]]
Out[2]= (E^(-((a - y z)^2/(2 s^2))) Abs[z])/(Sqrt[2 \[Pi]] s)
$Version
"8.0 for Microsoft Windows (64-bit) (October 7, 2011)"
Assuming[Element[a, Reals] && Element[y, Reals] && Element[z, Reals],
Integrate[
DiracDelta[y - (x + a)/z] PDF[NormalDistribution[0, s],
x], {x, -Infinity, Infinity}]]
(E^(-((a - y z)^2/(2 s^2))) Abs[z])/(Sqrt[2 \[Pi]] s)