16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Most important property of normal variables: any linear combination (weighted sum) of normal variables, whether independent or not, is another normal variable Note that for zero mean variables n2 f(x)= √2丌a p(1)=e o(t) to+f(x) Both are Gaussian forms The Normal Approximation to the Binomial Distribution The binomial distribution deals with the outcomes of n independent trials of an experiment. Thus if n is large, we should expect the binomial distribution to be well approximated by the normal distribution. The approximation is given by the normal distribution having the same mean and variance. Thus b(k, n, p znpq Relative error is of the order of (-np) (npg). The relative fit is good near the mean if npq is large and degenerates in the tails where the probability itself is small The Normal approximation to the poisson distribution Also the Poisson distribution depends on the outcomes of independent events. If 9/30/2004955AM Page 6 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 6 of 10 Most important property of normal variables: any linear combination (weighted sum) of normal variables, whether independent or not, is another normal variable. Note that for zero mean variables 2 2 2 2 2 2 1 ( ) 2 ( ) x t fx e t e σ σ πσ φ − − = = Both are Gaussian forms. The Normal Approximation to the Binomial Distribution The binomial distribution deals with the outcomes of n independent trials of an experiment. Thus if n is large, we should expect the binomial distribution to be well approximated by the normal distribution. The approximation is given by the normal distribution having the same mean and variance. Thus 2 ( ) 1 2 (,, ) 2 k np npq bkn p e πnpq − − ≈ Relative error is of the order of 3 2 ( ) ( ) k np npq − The relative fit is good near the mean if npq is large and degenerates in the tails where the probability itself is small. The Normal Approximation to the Poisson Distribution Also the Poisson distribution depends on the outcomes of independent events. If there are enough of them