expectation, the expected or mean\[ E\left [ X\right ] \equiv \mu _x=\int _{-\infty }^\infty x\;f_x\left ( x\right ) \;dx \]
if \(x\) is discrete then\[ E\left [ X\right ] \equiv \mu _x=\sum _nx_nP_x\left ( x_n\right ) \]
for a normalized system, i.e. total weight = 1, then \(\mu _x\) can be considered to be the center of gravity.
expected value of \(Y=G\left ( x\right ) \)
\[ E\left [ g\left ( x\right ) \right ] \equiv \mu _y=\int _{-\infty }^\infty y\;f_Y\left ( y\right ) \;dy=\int _{-\infty }^\infty g\left ( x\right ) f_Y\left ( y\right ) \;dy \]
\[ \mu _y=\int _{-\infty }^\infty g\left ( x\right ) f_X\left ( x\right ) \;dx \]
theorm\[ f_Y\left ( y\right ) =\frac{f_X\left ( x=g^{-1}\left ( y\right ) \right ) }{\left | g^{^{\prime }}\left ( x\right ) \right | } \]
\[ dy=\left | g^{^{\prime }}\left ( x\right ) \right | \;dx \]
conditonal expectation\[ E\left [ Y|B\right ] \equiv \int _{-\infty }^\infty y\;f_{Y|B}\left ( y|b\right ) \;dy \]
moments, nth moment
\(n^{th}\) moment of \(X\) denoted by \(\epsilon _n\)\[ \epsilon _n\equiv \int _{-\infty }^\infty x^n\;f_X\left ( x\right ) \;dx \]
\[ \begin{array}{l} \epsilon _0=1 \\ \epsilon _1=\mu _x\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \text{mean value}\; \\ \epsilon _2=E\left [ X^2\right ] \;\;\;\;\;\;\;\;\;\;\; \text{mean squared value}\end{array} \]
central moments
\[ m_n\equiv \int _{-\infty }^\infty \left ( u-\mu _x\right ) ^n\;f_X\left ( x\right ) \;dx \]
\[ \begin{array}{lll} m_0 & = & 1 \\ m_1^{} & = & 0 \\ m_2 & = & E\left [ \{X-\mu _X\}^2\right ] \;\;\;\;\;\;\;\;\; \text{spread or variance =}\sigma _x^2 \\ m_3 & = & E\left [ \{X-\mu _X\}^3\right ] \;\;\;\;\;\;\;\;\;\text{skew}\end{array} \]
standard deviation
\[ \sigma _x=\sqrt{m_2} \]
realtionships between moments
\[ \sigma ^2=m_2=\epsilon _2-\epsilon _1^2=E\left [ X^2\right ] -\left \{ E\left [ X\right ] \right \} ^2 \]
mean sequence
\[ \mu _X\left ( n\right ) \equiv E\left [ X_n\right ] =\int _{-\infty }^\infty x_n\;f\left ( x_n\right ) \;dx_n \]
autocorrelation Bisequence\[ R_X\left ( m,n\right ) \equiv E\left [ X_mX_n^{*}\right ] =\int \int x_mx_n^{*}\;f\left ( x_m,x_n\right ) \;dx_mdx_n \]
Auto covariance Bisequence\[ K_X\left ( m,n\right ) \equiv E\left [ \left \{ X_m-\mu _X\left ( m\right ) \right \} \left \{ X_m^{*}-\mu _X^{*}\left ( m\right ) \right \} \right ] \]
relation\[ K_X\left ( m,n\right ) =R_X\left ( m,n\right ) -\mu _X\left ( m\right ) \mu _X^{*}\left ( n\right ) \]
definitions
uncorrelated random sequence
\[ \begin{array}{l} if \\ K_x(m,n)=0\;\;\;\;\;\;\;\;\;\forall m,n\;\;\;m\neq n \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;=\sigma _x^2\;\;\;\;\;\;\;\;m=n\end{array} \]
or\[{\bf R_X\left ( m,n\right ) =\mu _X\left ( m\right ) \mu _X^{*}\left ( n\right ) \;\;\;\;\;\;\;\forall m,n\;\;\;m\neq n\;} \]
then the sequence is called uncorrelated random sequence
orthogonal random sequence
if\[ \begin{array}{l} R_X\left ( m,n\right ) =0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall m,n\;\;\;m\neq n \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=E\left [ x_n^2\right ] \;\;\;\;\;\;\;\;\;\;\;\;m=n\end{array} \]
then the sequence is called an orthogonal random sequence
Gausian random sequence
if all kth order distributions of a random sequence \(X_n\) are jointly Gaussian then it is called a Gaussian random seq.
strict sense stationary SSS
if the kth order probability functions do not depend on the index n, then it is SSS.
Wide sense stationary WSS
if the mean function is constant and the autocorrelation (covariance) is shift-invariant then it is WSS.
i.e.\[ \mu _x\left ( n\right ) =\mu _x \]
and\[ R_X\left ( m,n\right ) =R_X\left ( m-n\right ) \]
\(^{}\)
usefull identity\[ \int _0^\infty x^ne^{-x}\;dx=n! \]
when adding 2 i.i.d R.V., their f’s convolve and their characterstic functions is multiplied
if X is an i.i.d, then at each n it is the same RV, and they are independent RV’s
convolution
for a discrete, linear time invariant
\[ y\left ( n\right ) =\sum _{m=-\infty }^\infty h(n-m)u(m) \]
\[ y=h*u \]
for a continouse liner time invariant \[ y\left ( t\right ) =\int _{-\infty }^\infty h\left ( t-\tau \right ) u\left ( \tau \right ) \;d\tau \]
\[ y=h*u \]