I’m trying to plot a graph for the following expectation

$ $ \mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=a 2^{-\frac{\kappa }{2}-1} b^{-\frac{\kappa }{2}} \theta ^{-\kappa } \left(\frac{\, _2F_2\left(\frac{\kappa }{2}+\frac{1}{2},\frac{\kappa }{2};\frac{1}{2},\frac{\kappa }{2}+1;\frac{1}{2 b \theta ^2}\right)}{\Gamma \left(\frac{\kappa }{2}+1\right)}-\frac{\kappa \, _2F_2\left(\frac{\kappa }{2}+\frac{1}{2},\frac{\kappa }{2}+1;\frac{3}{2},\frac{\kappa }{2}+\frac{3}{2};\frac{1}{2 b \theta ^2}\right)}{\sqrt{2} \sqrt{b} \theta \Gamma \left(\frac{\kappa +3}{2}\right)}\right)$ $ where $ a$ and $ b$ are constant values, $ \mathcal{Q}$ is the Gaussian Q-function, which is defined as $ \mathcal{Q}(x) = \frac{1}{\sqrt{2 \pi}}\int_{x}^{\infty} e^{-u^2/2}du$ and $ \gamma$ is a random variable with Gamma distribition, i.e., $ f_{\gamma}(y) \sim \frac{1}{\Gamma(\kappa)\theta^{\kappa}} y^{\kappa-1} e^{-y/\theta} $ with $ \kappa > 0$ and $ \theta > 0$ .

This equation was also found with Mathematica, so it seems to be correct. I’ve got the same plotting issue with Matlab.

Follows some examples, where I have checked the analytical results against the simulated ones.

When $ \kappa = 12.85$ , $ \theta = 0.533397$ , $ a=3$ and $ b = 1/5$ it returns the correct value $ 0.0218116$ .

When $ \kappa = 12.85$ , $ \theta = 0.475391$ , $ a=3$ and $ b = 1/5$ it returns the correct value $ 0.0408816$ .

When $ \kappa = 12.85$ , $ \theta = 0.423692$ , $ a=3$ and $ b = 1/5$ it returns the value $ -1.49831$ , which is negative. However, the correct result should be a value around $ 0.0585$ .

When $ \kappa = 12.85$ , $ \theta = 0.336551$ , $ a=3$ and $ b = 1/5$ it returns the value $ 630902$ . However, the correct result should be a value around $ 0.1277$ .

Therefore, the issue happens as $ \theta$ decreases. For values of $ \theta > 0.423692$ the analytical matches the simulated results. The issue only happens when $ \theta <= 0.423692$ .

I’d like to know if that is an accuracy issue or if I’m missing something here and if there is a way to correctly plot a graph that matches the simulation. Perhaps there is another way to derive the above equation with other functions or there might be a way to simplify it and get more accurate results.