16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde P(k,u Relative error is of the order of (k-a) The relative fit is subject to the same behavior as the binomial approximation Interpretation of a continuous distribution approximating a discrete one: The value of the normal density function at any k approximates the value of the discrete distribution for that value of k. Think of spreading the area of each impulse over a unit interval. Then the height of each rectangle is the probability that the corresponding value of k will be taken. The normal curve approximate this step-wise function P(k=x) 012345 Note that in summing the probabilities for values of k in some interval, the approximating normal curve should be integrated over that interval plus 2 on each end to get all the probability associated with those values of k P(N≤X≤N2)=∑P(k) P(k,u) d Multidimensional normal Distribution Probability density function f(x)= exp[ -(x-DTM-(x-Y) 9/30/2004955AM Page 7 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 7 of 10 2 ( ) 1 2 (, ) 2 k Pk e µ µ µ πµ − − ≈ Relative error is of the order of 3 2 ( ) k µ µ − The relative fit is subject to the same behavior as the binomial approximation. Interpretation of a continuous distribution approximating a discrete one: The value of the normal density function at any k approximates the value of the discrete distribution for that value of k. Think of spreading the area of each impulse over a unit interval. Then the height of each rectangle is the probability that the corresponding value of k will be taken. The normal curve approximates this step-wise function. Note that in summing the probabilities for values of k in some interval, the approximating normal curve should be integrated over that interval plus ½ on each end to get all the probability associated with those values of k. 2 1 1 2 ( ) () N k N PN X N Pk = ≤≤ = ∑ 2 2 2 1 1 1 ( ) 2 2 1 2 2 1 1 (, ) 2 1 1 2 2 N N x k N N P k e dx N N µ µ µ πµ µ µ µ µ + − − = − ≈ ⎛ ⎞⎛ ⎞ +− −− ⎜ ⎟⎜ ⎟ =Φ −Φ ⎝ ⎠⎝ ⎠ ∑ ∫ Multidimensional Normal Distribution Probability density function: ( ) 1 2 1 1 ( ) exp ( ) ( ) 2 2 n f x xXM xX π M ⎡ ⎤ Τ − = −− − ⎢ ⎥ ⎣ ⎦