importantProperty.tex

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\begin{document}
    \section*{Page Break Equation}
    \begingroup
    \allowdisplaybreaks
    \begin{align*}
        \mathbb{P}(Y(t) = k)
        &= \sum_{i=0}^\infty \mathbb{P}(X(t) = k+i)\mathbb{P}(Y(t) = k\, | \, X(t)=k+i)\\
        &= \sum_{i=0}^\infty \frac{\lambda^{k+i}}{(k+i)!}e^{-\lambda}\binom{k+i}{k}p^k(1-p)^i\\
        &= \sum_{i=0}^\infty \frac{\lambda^k \lambda^i}{k!i!}e^{-\lambda}p^k(1-p)^i\\
        &= \frac{(p\lambda)^k}{k!}e^{-\lambda} \sum_{i=0}^\infty \frac{((1-p)\lambda)^i}{i!}\\
        &= \frac{(p\lambda)^k}{k!}e^{-\lambda}e^{(1-p)\lambda}\\
        &= \frac{(p\lambda)^k}{k!}e^{-p\lambda}.\\
    \end{align*}
    \endgroup
    %
    \section*{Breaking Equation in aligned line}
    \begin{align*}
        f_{T_n}(t)
        &= -\left(\lambda + \lambda^2t+\cdots+\frac{\lambda^{n-2}t^{n-3}}{(n-3)!}+\frac{\lambda^{n-1}t^{n-2}}{(n-2)!}\right)e^{-\lambda t}\notag \\
        & \quad \quad + \left(1 + \lambda t + \frac{(\lambda^2t)}{2!}+\cdots+\frac{(\lambda t)^{n-2}}{(n-2)!}+\frac{(\lambda t)^{n-1}}{(n-1)!}\right)\lambda e^{-\lambda t}\\
        &= \frac{\lambda^nt^{n-1}}{(n-1)!}e^{-\lambda t}
    \end{align*}
    %
    \section*{tfrac,frac,dfrac}
    \begin{align*}
        \mathbb{P}(A(\tfrac{3}{2})= 3, B(\tfrac{3}{2}) = 3\, | \, A(\tfrac{3}{4}) = 2, B(\tfrac{3}{4})=1)\\
        \mathbb{P}(A(\dfrac{3}{2})= 3, B(\dfrac{3}{2}) = 3\, | \, A(\dfrac{3}{4}) = 2, B(\dfrac{3}{4})=1)\\
        \mathbb{P}(A(\frac{3}{2})= 3, B(\frac{3}{2}) = 3\, | \, A(\frac{3}{4}) = 2, B(\frac{3}{4})=1)
    \end{align*}
\end{document}