Integration of multiplication of normally distributed random variables

Question

Assume we have three random variables X, Y, Z in which Y and Z are independent. What is the result of this integration ($N$ represents normal distribution over random variable with mean and variance) $$\int_Z\int_{Y} N(X|Y-Z,1) N(Z|\mu_z,\sigma_z^2)N(Y|\mu_y,\sigma_y^2) dydz$$ I know that $\int_{Y} N(X|Y,1) N(Y|\mu_y,\sigma_y^2) dy = N(X|\mu_y,\sigma_y+1)$ But the inclusion of Z into formula as $ N(X|Y-Z,1)$ makes this problem unsolvable for me.

Answer 1

This is not an answer but just an extended comment to give the associated Mathematica code.

result = Integrate[PDF[NormalDistribution[y - z, σx], X]* 
  PDF[NormalDistribution[μz, σz], z]*PDF[NormalDistribution[μy, σy], y],
  {z, -∞, ∞}, {y, -∞, ∞}, Assumptions -> {σx > 0, σy > 0, σz > 0}] 

One can make this look in a more usual form:

result /. {σx -> Subscript[σ, x], σy -> Subscript[σ, y], σz -> Subscript[σ, z],
 μy -> Subscript[μ, y], μz -> Subscript[μ, z]}