sigma := 10Ezero := 1000. ezeroprime := Ezero  -  eprime

\(\(
pgauss := \ 
      1/\((sigma*Sqrt[2*Pi])\)\ 
        Exp[\(-0.5\)*\((\((Energy - ezeroprime)\)/sigma)\)^2]\)\)

Plot[pgauss, {Energy, 950, 1050}, PlotPoints -> 100]

lambda :=  .5*(eprime  -  2)

plandau  :=  
      1/(2 Pi)  Exp[-(lambda  +  Exp[-lambda])]

Plot[plandau,  {eprime, 0. , 15}]

ptotal := NIntegrate[plandau*pgauss , {eprime,  .1, 20}]

FindMinimum[-ptotal, {Energy, 992, 1000}, AccuracyGoal -> 10000000]

expect := NIntegrate[eprime*plandau, {eprime, -1, 20}]

norm := NIntegrate[plandau, {eprime, -1, 20}]

N[expect/norm]