In the following µ represents lambda.
Marginal probability distributions are given by summations of the same function but different limits.
For x: P(X=x)=e^(-µ)∑((µ/2)^x)/(y!(x-y)!) with limits {low,high}={y=0,y=x} because x-y≥0 so that the factorial argument≥0. Example: if x=10, the limits are {0,10}.
For y: P(Y=y)=e^(-µ)∑((µ/2)^x)/(y!(x-y)!) with limits {x=y,x=∞}. Example: if y=10, the limits are {10,∞}.
I hope this answer helps.