Calculating probability on a given non-integer range interval for poisson distribution

Question

I am bit confused on how to calculate probability in a given range for r.v. given poison distribution.

I wanted to calculate $P( 0.5 < x \leq 1.5 )$ given Poisson$(2)$.

In most books they only show X = x and not in ranges like the one above.

Should I calculate $P(x) = 1 - F(1.5) - F(0.5)$ ?

where $F(x)$ is $\frac{ lambda^x e**(-lambda) }{ x! } $

Problem here also is that I have float ranges and not integers, how can this be calculated?

Answer 1

A Poisson-distributed random variable can only take on nonnegative integer values; e.g., if $$X \sim \operatorname{Poisson}(\lambda), \\ \Pr[X = x] = e^{-\lambda} \frac{\lambda^x}{x!}, \quad x \in \{0, 1, 2, \ldots \},$$ then $X$ can never be non-integer or negative. So $$\Pr[0.5 < X \le 1.5] = \Pr[X = 1],$$ because there is no other outcome for $X$ that satisfies the inequality $0.5 < X \le 1.5$.

Answer 2

The Poisson is a discrete law, with support

$x=0,1,2,3,...$

Thus

$$P(0.5<X\leq1.5)=P(X=1)$$

Note: respect to your previous question, here between $<$ and $\leq$ there is a big difference