16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde A very general quantitative statement of this is known as Chebyshevs P(X-X1)≤ This is a very general result in that it holds for any distribution of X. lote that it gives you just an upper bound on the probability of exceeding a this bound to be a tight one quantitativel/p lete general, we should not expect given deviation from the mean. Being com powerful analytical tool. For quantitative practical work it is preferable if possible to estimate the distribution of X and use this to calculate the probability indicated above Characteristic Function 1. Definition of Characteristic Function 2. Inverse relation 3. Characteristic function and inverse for discrete distributions 4. Prove o(1)=∏中() where X=X1+X2+…+Xn( Y, independent) 5. MacLaurin series expansion of p(t) 1. Definition of Characteristic function p(1)=E(e") Iefr(x)dx The characteristic function of a random variable X with density functionf(x) defined to be the expectation of e/ the integral expression then follows from the relation for the expectation of a function of a random variable The main purpose in defining the characteristic function is its utility in the treatment of many problems 2. Inverse relation The expression for o(t) is in the form of the inverse Fourier transform of the density function fo X; thus by the theory of Fourier transforms the direct relation must hold and the functions involved are uniquely related note that the integral always converges Page 6 of 716.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde A very general quantitative statement of this is known as Chebyshev’s Inequality: 2 σ PX (| − X |≥ t) ≤ t 2 This is a very general result in that it holds for any distribution of X. Note that it gives you just an upper bound on the probability of exceeding a given deviation from the mean. Being complete general, we should not expect this bound to be a tight one quantitatively. Because of its generality, it is a powerful analytical tool. For quantitative practical work it is preferable if possible to estimate the distribution of X and use this to calculate the probability indicated above. Characteristic Function 1. Definition of Characteristic Function 2. Inverse relation 3. Characteristic function and inverse for discrete distributions 4. Prove φ ( )t t = Πφ i ( ) where X = X1 + X 2 + ... + X (Xi independent) x x n 5. MacLaurin series expansion of φ(t) 1. Definition of Characteristic Function jtx φ( )t = E e ( ) ∞ ef ( ) jtx = x dx ∫ x −∞ The characteristic function of a random variable X with density function fx(x) is defined to be the expectation of e jtx; the integral expression then follows from the relation for the expectation of a function of a random variable. The main purpose in defining the characteristic function is its utility in the treatment of many problems. 2. Inverse relation The expression for φ(t) is in the form of the inverse Fourier transform of the density function fo X; thus by the theory of Fourier transforms the direct relation must hold and the functions involved are uniquely related. Note that the integral always converges. Page 6 of 7