Poisson PDF with non-integer support

[Oxymoron]

Homework Statement

If $$X$$ is a Poisson random variable with $$\lambda = 2$$ find the probability that $$X>0.5$$.

Homework Equations

The Poisson PDF:
$$P(x,\lambda) = \frac{\lambda^k}{k!}e^{-\lambda}$$

The Attempt at a Solution

Usually with these sorts of probability problems where they ask you to find the probability that $$x$$ is larger than some number $$n$$ I use the CDF of the PDF and write

$$P(X_{PDF}>n) = 1-P(X_{PDF}\leq n) = 1-P(X_{CDF}=n)$$

However, I am at a loss with the Poisson distribution because the CDF involves the gamma function. I can do it on Maple where I define

$$\mbox{Poi}(\lambda,x) := \sum_{t=0}^x \frac{\lambda^t}{t!}e^{-\lambda}$$

and then calculate

$$1-\mbox{evalf}(\mbox{Poi}(2,0.5)) = 0.7385...$$

Also, if I try to use z-scores in a Poisson table the values for x are all integers, am I meant to use interpolation? Or is there an algebraic way of solving this?

 

[Oxymoron]

Solved.

I used the cumulative distribution function for Poisson:

$$F(t,\lambda) = \frac{\Gamma\left(\lfloor k+1 \rfloor,\lambda\right)}{\lfloor k \rfloor!}$$

and used the incomplete gamma function

$$\Gamma(k,x) = \int_x^{\infty}t^{k-1}e^t\mbox{d}t$$

and integrated by parts twice (twice because the support is $$\lambda = 2$$ by the way!) to find an answer. It turns out that non-integers can be put into the gamma function, but it just floors them anyway. Did it on Maple as well as by hand and it works.