Univariate normal distribution with mean μ and variance (σ^2):

f[x_,μ_,σ_]=1/(σ Sqrt[2Pi])Exp[-(x-μ)^2/(2 (σ^2))]

Bivariate joint density:

Σ={{1,ρ},{ρ,1}};Σ//MatrixForm

ff[x1_,x2_,μ1_,μ2_,ρ_]=1/(2Pi Sqrt[ 1-ρ^2])Exp[-((x1-μ1)^2-2ρ(x1-μ1)(x2-μ2)+(x2-μ2)^2)/(2(1-ρ^2))]

Conditional distribution

denom=Assuming[ρ<1 && ρ>-1,Integrate[ff[y,x2,μ1,μ2,ρ],{y,-Infinity,Infinity}]]

Simplify[ff[x1,x2,μ1,μ2,ρ]/denom]