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

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

Prof vander Velde Use error between early and late indicator to lock onto signal. Error is a linear function of shift, within the range (-T,T) Return to the 1 st example process and take the case where the change points are Poisson distributed

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

Mean squared value: Higher order distribution and density functions. You can define these distributions of any order

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

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

The Poisson approximation to the binomial distribution The binomial distribution, like the Poisson, is that of a random variable taking only positive integral values. Since it involves factorials, the binomial distribution is not very convenient for numerical application

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