# [Math] My Math Note

Taken somewhere from Internet.

Variance, covariance and autocovariance:

• Variance is one of the parameter to describe the distribution. It tells how the distribution extends from the mean value.
• Moments are always used to describe the distribution due to its computational simplicity (i.e. the power series). 1st moment is mean value, 2nd moment is variance, 3rd moment is skewness.
• Let $X$ and $Y$ denote two random variables. Covariance $COV(X,Y)$ is a measure of how much $X$ and $Y$ change together.
• In this case, variance is a special case of the covariance when two variables are identical.
• Given a stochastic process $X(t)$, the autocovariance is the covariance of $X(t)$ againsts itself in a time-shifting. That is $COV(X(t), X(t-\tau))$.

Correlation and Autocorrelation:

• Correlation is the measure of the strength of association between two variables. For example, there is a strong correlation between black eyes and and black hair as you may seem them moving together all the time. But, there is a weak correlation between blue wallpaper and puppies. Bear in mind that correlation is not causation. Though the death of babies increase with the cases of vehicle accidents. They have a strong correlation. But we cannot say the death of babies causes the happen of vehicle accidents. It does not make sense. [Taken from Yahoo! Answer]
• Autocorrelation is the cross-correlation of the variable itself, in which separated by certain amount of time delay.

Random Variable:

• Say we run an experiment in which it gives a finite number of possible outcome, named as sample space $S$.
• For the convenience of mathematical calculation, each outcome (denoted as $\zeta$) of $S$ is translated into a real number (if possible) and that translation process is given by a function $X(\zeta)$ where $\zeta \in S$. We name $X(\zeta)$ as random variable and most of the time we just denote it as $X$ if the context is well understood.

Stationary Random Process:

• Let $X={x(t)}$ denote a random process, in which $u_x(t)$ and $R_{xx}(t,t+\tau)$ denote its mean value and autocorrelation function.
• $X$ is non-stationary if $u_x(t)$ and $R_{xx}(t,t+\tau)$ vary as $t$ varies.
• Otherwise, $X$ is said stationary or weakly stationary in wide-sense.
• For weakly stationary process, $u_x(t)=u_x$ and $R_{xx}(t,t+\tau)=R_{xx}(\tau)$.
• For strongly stationary process, all possible moments and join moments are time-invariant.