Direct determination of the joint probability density of several functions o several ra andom variables Suppose we have the joint probability density function of several random variables x, Y, Z, and we wish the joint density of several other random variables defined as functions xyz

System parameters are contained in,(t) and(t) Desired output is generated by taking the signal through the desired operator. The difference between the actual output and the desired output is the error, whose mean squared value we want to minimize We require a stable and realizable system. The error e(t) is given by:

Note that in the case of repeated roots of the denominator, a pole of multiple order contributes only a single residue. To evaluate F(s)ds by integrating around a closed contour enclosing the entire left half plane, note that if F()0 faster than-for large s, the integral along the curved part of the contour is zero

This is the main reason why use of the characteristic function is convenient This would also follow from the more devious reasoning of the density function for the sum of n independent random variables being the nth order convolution of the individual density functions-and the knowledge that convolution in the direct variable domain becomes multiplication in the transform domain

16.322 Stochastic Estimation and Control Professor Vander Velde 1. P(ABCD.=P(A)P(B A)P(C|AB)P(D 1 ABC) Derive this by letting A=CD. Then P(BCD)= P(CD)P(B ICD)= P(C)P(DIC)P(DICD) 2. If A,, A2r.. is a set of mutually exclusive and collectively exhaustive events, then