How To Find the Probability of a Gamma Distributed Random Variable?

Suppose a random variable has a gamma distribution with $$\alpha = 0.8$$ and $$\beta = 2.4$$

How can we calculate $$P(Y > 3)$$?

My book says when $$d$$ and $$c$$ are such that $$0 < c < d < \infty$$ the integral of the gamma pdf cannot be directly integrated:

$$\int_{c}^{d}\frac{{y^{\alpha-1}e^{-y/\beta}}}{\beta^\alpha\Gamma(\alpha)}dy$$

But for $$P(Y > 3)$$ would I want to calculate the following? $$1-\int_{0}^{3}\frac{{y^{-.2}e^{-y/2.4}}}{2.4^.8\Gamma(.8)}dy$$ Since in this case $$0=c. Does that make things easier?

Also, for $$\Gamma(.8)$$, wolfram gives me $$1.16$$ but my TI-84 gives me $$0$$? Does anyone know how to do it correctly on the calculator?

• Correct, the survival function is given by the second formula ($1 - \int_0^3$). It's the incomplete gamma function. I suppose you'll have to implement your own numerical routine on the calculator for $\Gamma(x), \, 2 x \not \in \mathbb N$. – Maxim Apr 2 at 19:15