Limit theorem Chebyshev’s inequality \begin{align} {\sigma ^2} &= \int {{{\left( {x – \mu } \right)}^2}{f_X}\left( x \right)dx} \\ &\ge \int_{ – \infty }^{u – c} {{{\left( {x – \mu } \right)}^2}{f_X}\left( x \right)dx} + \int_{u + c}^\infty {{{\left( {x – \mu } \right)}^2}{f_X}\left( x \right)dx} \\ &\ge {c^2}\int_{ – \infty }^{u – c} {{f_X}\left( x \right)dx} {\rm{ + }}{c^2}\int_{u + c}^\infty {{f_X}\left( x \right)dx} \\ &={c^2}{\rm{P}}\left( {\left| {X – \mu } \right| \ge c} \right) \end{align} \[{\rm{P}}\left( {\left| {X – …
Category: Notes
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 = …
Independent and disjoint Two events are independent or disjoint are two different concept. If two events are independent, there is no relation between but they can happen together. It is noted that independence can be affected by conditioning. If two events are disjoint, they has the relation that when one happens the other cannot happen and they are not independent. Independent: \({\rm{P}}\left( {B\left| A \right.} \right) = {\rm{P}}\left( B \right) \Leftrightarrow {\rm{P}}\left( {A\left| B \right.} \right) = {\rm{P}}\left( A \right)\) Disjoint: \({\rm{P}}\left( …
ODE system \[\left\{ \begin{array}{l} {x^{\prime}} = f\left( {x,y,t} \right)\\ {y^{\prime}} = g\left( {x,y,t} \right) \end{array} \right.,x\left( {{t_0}} \right) = {x_0},y\left( {{t_0}} \right) = {y_0}\] Linear system: \[\left\{ \begin{array}{l} {x^{\prime}} = ax + by + {r_1}\left( t \right)\\ {y^{\prime}} = cx + dy + {r_2}\left( t \right) \end{array} \right.\] \(a\), \(b\), \(c\), \(d\) can be functions of \(t\). If they are constant, then the system is called constant coefficient system. When \(r_1\left( t \right) = {r_2}\left( t \right) = 0\), the system is linearly homogeneous. Example: …
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, …