bin_poiss - The binomial and Poisson distributions There is...

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The binomial and Poisson distributions There is a short list of probability distributions with names which crop up in some speciFc engineering problems. We will look at just a few as they’re important but not the central theme of this set of lectures. Two of them, the binomial and Poisson distributions are relevant to the problem of counting random events . These two pages are meant to complement the brief treatment in § 23.6 of Kreyszig. The binomial distribution is relevant when counting the total number of ‘failures’ in a Fxed length series of trials, each of which is independently a ‘failure’ with probability p and otherwise a ‘success’. Suppose there are n trials and let T denote the total number of failures. Clearly T can only take values from 0 up to n . We can write T = F 1 + F 2 + . . . + F n , where each F i is an independent random variable with P [ F i = 1] = p and P [ F i = 0] = 1 - p , (so the F i tell us which trials the failures are in while T just counts the number of them – it is not strictly necessary to introduce the F i here but I think they are natural and they will be useful later). Now, by extending the notion that e.g. P [ F 1 = 1 and F 2 = 0] = P [ F 1 = 1] × P [ F 2 = 0] = p (1 - p ) to greater numbers of independent random variables we can say P [ F 1 = 1 , . . . , F t = 1 , F t +1 = 0
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bin_poiss - The binomial and Poisson distributions There is...

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