which we define as the correlation. Often we do not know the complete distribution, but only simple statistics. The most common of the moments of higher ordered distribution functions is the covarance

In the special case when E's are conditionally independent(though they all depend on the alternative, Ak), P(, E2..)= P(A... -)P() ()P) This is easy to do and can be done recursively

If a set of random variables X, having the multidimensional normal distribution is uncorrelated(the covariance matrix is diagonal, they are independent. The argument of the exponential becomes the sum over i of Thus, the distribution becomes a product of exponential

Non-zero power at non-zero frequency If R(r) includes a sinusoidal component corresponding to the component x()=Asin(o41+6) where 0 is uniformly distributed over 2t, A is random independent of 0, that component will be

where is the linearized system matrix. But this requires the full(same number of equations as finite differencing). In =time when the nominal trajectory

Last time: Ergodic processes An ergodic process is necessarily stationary. Example: Binary process

so if t is the current time in a real time situation, we cannot compute ()for which is necessary since w, () is nonzero only for>0. But we showed earlier that

Note n, s enter the system at the same place F is fixed. We design(and perhaps B). We must stabilize F if it is given as unstable

Solution in the Free Configuration Non-Real-Time Filter Case The applicable condition in this important case is:

The two middle terms are zero: fort>lo,n(t) and are uncorrelated becauset) is white(impulse correlation function) For=, n() has a finite effect on x()because n() is white. But the integral of a finite quantity over one point is zero