16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde X=x d x 2 d x b3-a3 3 6 (a2+ab+b2) x2=(a2+ab+b2)-(a2+2ab+b2) (a2-2ab+b2) 12 O 12 The binomial distribution Outl 1. Definition of the distribution 2. Determination of the binomial coefficient and binomial distribution 3. Useful relations in dealing with binomial coefficients and factorials 4. The mean, mean square and variance of the binomial distribution 1. Definition of the distribution Consider an experiment in which we identify two outcomes: one of which we call success and the other failure. The conduct of this experiment and the observation of the outcome may be called a simple trial. If the trial is then repeated under such circumstances that we consider the outcome on any trial to be independent of the outcomes on all other trials, we have a process frequently called bernoulli Trials after the man who first studied at length the results of such a process The number of successes in n bernoulli trials is a random discrete variable whose distribution is known as the binomial distribution ote that the binomial distribution need not refer p s observing the number of heads in n tosses of a coin. An experiment may have a great many simple outcomes- the outcomes may even be continuously Page 5 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 1 2 − 2 b X = ∫ x 1 dx = 2 (b a ) ba − b a − a 1 = (a b + ) 2 b X 2 1 1 3 = ∫ x 2 1 dx = (b a 3 − ) ba − − 3 b a a 2 = 1 (a ab b 2 + + ) 3 2 X 2 2 σ = − = 1 (a ab b 2 ) 2ab b 2 X ) 2 + + − 1 (a 2 + + 3 4 = 1 (a 2 − 2ab b + 2 ) 12 = 1 (b a) 2 − 12 1 σ = (b a − ) 12 The Binomial Distribution Outline: 1. Definition of the distribution 2. Determination of the binomial coefficient and binomial distribution 3. Useful relations in dealing with binomial coefficients and factorials 4. The mean, mean square, and variance of the binomial distribution 1. Definition of the distribution Consider an experiment in which we identify two outcomes: one of which we call success and the other failure. The conduct of this experiment and the observation of the outcome may be called a simple trial. If the trial is then repeated under such circumstances that we consider the outcome on any trial to be independent of the outcomes on all other trials, we have a process frequently called Bernoulli Trials after the man who first studied at length the results of such a process. The number of successes in n Bernoulli trials is a random discrete variable whose distribution is known as the Binomial Distribution. Note that the binomial distribution need not refer only to such simple situations as observing the number of heads in n tosses of a coin. An experiment may have a great many simple outcomes – the outcomes may even be continuously Page 5 of 10