CDF of SND
\begin{align*}
\Phi(a) &= 1 - \Phi(-a) \\
\Phi(-a) &= 1 - \Phi(a)
\end{align*}
\begin{gather*}
\because \Phi(-a) = P(Z \le -a) = 1 - P(Z \gt -a) = 1 - P(Z \lt a) = 1 - \Phi(a)
\end{gather*}
NOTE: P(Z \lt a) = P(Z \le a) is satisfied for continuous distribution.
Integration calc.
\begin{gather*}
\displaystyle\int e^{-(\lambda - \theta) x} dx = \frac{1}{-(\lambda - \theta)} e^{-(\lambda - \theta) x} \\
\because u \colonequals -(\lambda - \theta) x \rArr \frac{du}{dx} = -(\lambda - \theta) \rArr dx = \frac{du}{-(\lambda - \theta)} \tag{1}
\end{gather*}
NOTE: (1): Integration by substitution.
Discussion