15.1.12 Theory of random processes

Problem: The random process \( X(t), t \geq 0 \), is defined by the formula \( X(t)=\alpha \cos (t+\beta)+\varepsilon \), where \( \alpha, \beta, \varepsilon- \) are independent random variables, moreover, \( \alpha \sim N(0,1), \varepsilon \sim N\left(0, \sigma^{2}\right), \beta \sim U[-\pi, \pi] \). Find: \( P\left(X\left(t_{1}\right) \leq X\left(t_{2}\right) \mid \alpha \geq 0\right) \), where \( 0 \leq t_{1} \leq t_{1} \leq \frac{\pi}{2} \).