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 – …

# Tag: Probability

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( …