maximum of conditional i.i.d random variables

Question

I have a set of random variables $Z_1, Z_2, Z_3, ..., Z_n$ where $Z_1=X_1+A+b$ (Z are conditioned on A and b is a constant). The set of random variables $X_1,X_2,..,X_n$ are conditioned on random variable A. I am trying to find the CDF associated with the maximum of these random variables: $ Z_{max}=max (Z_1,Z_2,...,Z_n )$. My process of thinking is the following: $F_{Z_{max}}(z)=\int_0^{\infty} F_{Z|A}^n(z|\alpha) f_A(\alpha) d\alpha $ where $F_{Z|A}(z|\alpha)=F_{Z}(x-\alpha)$ and $f_A(\alpha)$ is the pdf associated with the random variable A for example, let say $X$ and $A$ are both exponential with rates $\mu$ and $\lambda$ respectively, then $F_{Z_{max}}(z)=\int_b^{z} (1-e^{-\mu(z-\alpha)})^n \lambda e^{-\lambda (\alpha-b)} d\alpha $. My problem is when I cauclate the mean using numerical integration and using monte carlo simulation are not similar. What did I do wrong here?