Calculating mean of a Compound Poisson variable with shifted Poisson distribution

Question

Suppose we have a compound random variable $$Y=\sum_{i=1}^{N}X_i$$, where $N$ and $X_i$ are independent random variables. $N$ has Poisson distribution with mean $\lambda_A$ and each $X_i$ has a shifted Poisson distribution as follows; $$P_N(n)=\frac{\lambda_A^n}{n!}e^{-\lambda_A}, n=0,1,2,3,...$$

$$P_X(x)=\frac{\lambda_X^{x}}{x!}e^{-\lambda_X}, x=0,2,3,...$$

Then, the probability mass function of $Y$ should be as follows: $P_Y(y)=\sum_{n=0}^{x} \left(\frac{(n\lambda_X)^{(y-n)}}{(y-n)!}e^{-n\lambda_X}\right)P_N(n)$

I would appreciate it if someone helps me how to calculate $E\{Y\}$?

MATLAB codes for calculating E[Y]