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

RNA polymerase transcribes genes How is this process regulated? How is specificity achieved? Key players transcriptional activator proteins -bind to specific sites on DNA and turn on the expression of nearby genes. Two adhesive surfaces one binds DNA, the other interacts with, and recruits RNA polymerase

Muscle cells But if in the same organIsm, should have the same DNA per Nerve cells( cell Different cells have different clusters of genes active Development and differentiation of embryos is controlled by signaling molecules that alter gene expression in cells

ADCS Motivation Motivation Sensors GPS star trackers. limb In order to point and slew optical sensors rate gy ros. inertial systems, spacecraft attitude control measurement units provides coarse pointing while

Definition A real-valued function V: X H R defined on state space X of a system with behavior set B and state r:B×[0,∞)→ X is called a Lyapunov function if tHv(t)=v(a(t))=v(a(z(), t)) is a non-increasing function of time for every z E B according to this definition, Lyapunov

6. Choice Under Uncertainty a. Representing Uncertainty: Lotteries and Compound Lotteries b. Axioms of Expected Utility C. The Expected Utility Theory d. Empirical Challenges to Expected Utility Theory-the Paradox Business e. Application: Crime and punishment

This lecture presents results describing the relation between existence of Lyapunov or storage functions and stability of dynamical systems 6.1 Stability of an equilibria

3. Comparative Statics a. Indirect Utility Functions b. Expenditure Functions and duality C. Expenditure Function and Price Indices d. Slutsky via Utility Functions e. Slutsky via Preferences

Chapter 17-Learning Objectives Build polynomial regression models to describe curvilinear relationships Apply qualitative variables representing two or three categories Use logarithmic transforms in constructing exponential and multiplicative models Identify and compensate for multicollinearity Apply stepwise regression

Chapter 16 Learning Objectives Obtain and interpret the multiple regression equation Make estimates using the regression model: Point value of the dependent variable,y Intervals: Confidence interval for the conditional mean of y Prediction interval for an individual y observation