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)

Hi

MMA says:

   $Version
   (* "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)"*)

   Integrate[DiracDelta[y - (x + a)/z] PDF[NormalDistribution[0, s], x], {x, -Infinity, Infinity}, 
   Assumptions -> {{a, y, z} \[Element] Reals, s > 0}]
   (* (E^(-((a - y z)^2/(2 s^2))) Abs[z])/(Sqrt[2 \[Pi]] s) *)

If you don't have Mathematica 11.3,you can try for free,you must only sign-in.

Regards,MI