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

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

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

Summary of the subject (topics) 1. Brief review of probability a. Example applications 2. Brief review of random variables a. Example applications 3. Brief review of random processes a. Classical description b. State space description

SYSTEM TDENTIFICATION DEVELOPING AN APPR0 PR1ATE MOOEL OF A YWAMIC SYSTEM US(NG BSERVED DATA COMBINEO WITH: BASIC MECHANICS AND OYNAMICS PR1OR KNOWLEVGE oF RELATIONSKIPS BETWEEN SIGNALS INPUT/OUTPUT MOOELS

Bachelor of Engineering The State University of Buffalo, 2002 Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics

USuALLY WE NOT KNOw THE \ACTUAL SYSTE\ So How WE ESTABLISH IF OUR MODEL Is Goo0? VARIOUS TPES OF TESTS CAN BE PERFORMEO PREDICTION ANO SIMULATIONJ ERRORS FRERUENCY RESPONSE FIT >MAKE SVRE YOU USE O1FFERENT 4TA To