# Let $X\sim Poi(10)$ and $Y \sim exp(\frac{1}{10})$ independent. Why $P(X+Y\leq\frac{3}{2}) = P(X=0, Y \leq \frac{3}{2}) + P(X=1,Y\leq \frac {1}{2})$?

Let $$X\sim Poi(10)$$ and $$Y \sim exp(\frac{1}{10})$$ independent random raviables

I would like to compute: $$P(X+Y\leq\frac{3}{2})$$

So what I did, which is probably wrong, is the following:

Let $$X=k$$ then $$P(X+Y\leq\frac{3}{2})=P(X=k, Y\leq\frac{3}{2}-k)$$

But I don't know how to take it from there.

The solution on the other hand computes it as follows:

$$P(X+Y\leq\frac{3}{2}) = P(X=0, Y \leq \frac{3}{2}) + P(X=1,Y\leq \frac {1}{2})$$

But why does $$X\in$${0,1} ? it doesn't make much sense to me.

Thanks!!

$$X$$ and $$Y$$ take only non-negative integer values. $$X+Y \leq \frac 3 2$$ implies $$X\leq \frac 3 2$$ and the only integers less than or equal to $$\frac 3 2$$ are $$0$$ and $$1$$.
That's because $$X \in \mathbb{N} \cup \{0\}$$ and $$Y > 0$$, so $$X + Y$$ can only be below $$3/2$$ when $$X = 0$$ or $$X = 1$$.