## Probabilistic system analysis – Part II

Bernoulli process \begin{align} {\rm{P}}\left( {{\rm{sucess}}} \right) &= {\rm{P}}\left( {{X_i} = 1} \right) = p\\ {\rm{P}}\left( {{\rm{failure}}} \right) &= {\rm{P}}\left( {{X_i} = 0} \right) = 1 – p\\ {\rm{E}}\left[ {{X_t}} \right] &= p,{\mathop{\rm var}} \left( {{X_t}} \right) = p\left( {1 – p} \right) \end{align} PMF of # of arrivals (number of success $S$ in $n$ time slots): binomial pmf \begin{align} {\rm{P}}\left( {S = k} \right) &= \left( \begin{array}{l} n\\ k \end{array} \right){p^k}{\left( {1 – p} \right)^{n – k}}, \ k =

## Differential equation – Part III Fourier series

Fourier Series To solve the ODE ${y^{\prime\prime}} + a{y^{\prime}} + by = f\left( t \right)$, in which $f\left( t \right)$ is often the combination of ${e^t}$, $\sin t$ and $\cos t$, then any reasonable $f\left( t \right)$ which is periodic, can be discontinuous but not terribly discontinuous, can be written as Fourier Series. $f\left( t \right) = {c_0} + \sum\limits_{n = 1}^\infty {{a_n}\cos nt + {b_n}\sin nt}$ Theorem: $u\left( t \right),v\left( t \right)$ are functions with period $2\pi$ on the real number,

## Differential equation – Part II Second-Order ODE

Linear 2nd order ODE with constant coefficients Homogeneous: ${y^{\prime\prime}} + A{y^{\prime}} + By = 0$ Solution: $y = {c_1}{y_1} + {c_2}{y_2}$ in which, ${y_1}$ and ${y_2}$ are two independent solutions of the homogeneous ODE. The basic method to solve this ODE is to try $y = {e^{rt}}$. Plug in and we can get ${r^2}{e^{rt}} + Ar{e^{rt}} + B{e^{rt}} = 0 \to {r^2} + Ar + B = 0$ Case 1: two different real roots $y = {c_1}{e^{{r_1}t}} + {c_2}{e^{{r_2}t}}$ Case

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