I have a stochastic process $X$ that is distributed according to the Gamma distribution: $X$ ~ $\Gamma(\alpha, \beta)$
I would like to compute the following : Probability = $P(X_t > 0.75 | X_{t-1} > 0.75)$ knowing that $X_t$ and $X_{t-1} $ are not independent.
I know that the cumulative distribution function of the Gamma distribution can be approximated as:
$F(x; \alpha, \beta) = \frac{\gamma(\alpha, \frac{x}{\beta})}{\Gamma(\alpha)}$ where $\gamma(\alpha, \frac{x}{\beta})$ is the lower incomplete gamma function.
I am not looking for the closed form expression (since I can implement the computation using a statistical package such as $R$), but I am looking to express that probability in terms of $P(X_t > 0.75)$, $P(X_{t-1} > 0.75)$, $\alpha$, and $\beta$.
Thank you!