16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 8 ast time: Multi-dimensional normal distribution f(x)= exp 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 terms in i If XY=0 The general moment of a multidimensional normal distribution, E X1X,2.X is known Laning and Battin. Random Processes in Automatic Control The Exponential Distribution Many components, especially electronic components, display constant percentage failure rates over long intervals Relati N(△t) rate of Time Ignore the " burn-in Start the time Approximation period axis here breaks down here Relative rate of failure vs time The familiar bathtub curve Failure rate= E(relative rate of failure) USing the random variable Tfor time to failure, and assuming a constant failure rate a (independent of time n) implies P(t<T≤1+dt|7>1)=dt Page 1 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Lecture 8 Last time: Multi-dimensional normal distribution 1 x exp ⎢ ⎡ − 2 ( x − x ) M ( x − x ) ⎤ f () = ⎣ 1 T −1 ⎦ n ⎥ (2π ) 2 M If a set of random variables Xi having the multidimensional normal distribution is uncorrelated (the covariance matrix is diagonal), they are independent. The 2 argument of the exponential becomes the sum over i of xi . Thus, the 2 distribution becomes a product of exponential terms in i. If XY = 0 23 2 X Y = X Y 3 = 0 r 2 r n r The general moment of a multidimensional normal distribution, E 1 ⎡XX 2 ...Xn ⎦ ⎤, ⎣ 1 is known. • Laning and Battin. Random Processes in Automatic Control. The Exponential Distribution Many components, especially electronic components, display constant percentage failure rates over long intervals. Relative rate of failure vs time The familiar “bathtub” curve. Failure rate = E(relative rate of failure) Using the random variable T for time to failure, and assuming a constant failure rate λ (independent of time t) implies P t ( < ≤ T t + dt T | > = t) λdt Page 1 of 6