PDF[TransformedDistribution[x z, {x [Distributed] ExponentialDistribution[[Lambda]],
   z [Distributed] NormalDistribution[0, 1]}], y]

(* Generate some data *)
SeedRandom[12345];
[Lambda]0 = 3;
n = 100;
x = RandomVariate[ExponentialDistribution[[Lambda]0], n];
z = RandomVariate[NormalDistribution[], n];
y = x z;

(* Log likelihood *)
logL = n Log[[Lambda]] - (3 n/2) Log[2] - n Log[[Pi]] +
   Sum[Log[MeijerG[{{}, {}}, {{0, 0, 1/2}, {}}, (
      y[[i]]^2 [Lambda]^2)/8]], {i, n}];

(* Method of moments estimate *)
[Lambda]Initial = Sqrt[2]/StandardDeviation[y]
(* 3.159259259137787 *)

(* Maximum likelihood estimate *)
mle = FindMaximum[{logL, [Lambda] > 0}, {[Lambda], [Lambda]Initial}]
(* {33.75465530200371,{[Lambda] -> 2.7946001175877813}} *)
(* Alternative (but slower) *)
(* mle=NMaximize[{logL,[Lambda]>0},[Lambda],Method -> "SimulatedAnnealing"] *)

(* Check to see if first partial derivative is approximately zero *)
dlogL[Lambda] = D[logL, [Lambda]] /. mle[[2]]
(* 1.000782830828939 * 10^(-6) *)

(* Estimate of standard error *)
se = Sqrt[-1/(D[logL, {[Lambda], 2}]) /. mle[[2]]]
(* 0.39489891632730545 *)

(* Another check:  plot the log likelihood *)
Plot[logL, {[Lambda], 1, 4}, Frame -> True, FrameLabel -> {"[Lambda]", "Log(likelihood)"}]