The entries in the table are the probabilities P X k of individual outcomes The

# The entries in the table are the probabilities p x k

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book. The entries in the table are the probabilities P ( X = k ) of individual outcomes. The values of p that appear in Table C are all 0.5 or smaller. When the probability of a success is greater than 0.5, restate the problem in terms of the number of failures.

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Normal approximation If n is large, and p is not too close to 0 or 1, the binomial distribution can be approximated by the normal distribution N ( µ = np, σ 2 = np (1 p )) Practically, the Normal approximation can be used when both np ≥10 and n (1 p ) ≥10. If X is the count of successes in the sample, the sampling distributions for large n is: X approximately N ( µ = np, σ 2 = np (1 p ))
Binomial mean and standard deviation The center and spread of the binomial distribution for a count X are defined by the mean µ and standard deviation σ : ) 1 ( p np npq np = = = σ µ Effect of changing p when n is fixed. a) n = 10, p = 0.25 b) n = 10, p = 0.5 c) n = 10, p = 0.75 For small samples, binomial distributions are skewed when p is different from 0.5. 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 Number of successes P(X=x) 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 Number of successes P(X=x) 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 Number of successes P(X=x) a) b) c)

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The death of Prussian Cavalrymen Suppose you, like von Bortkiewicz, are interested in the probability that Cavalrymen are killed by a horse kick. For 20 years, he studied 10 corps of the Prussian army giving him 200 observations. The total deaths from horse kicks were 122, and the average number of deaths per year per corps was thus 122/200 = 0.61. This is a rate of less than 1. It is also obvious that it is meaningless to ask how many times per year a cavalryman was not killed by the kick of a horse. In any given year, we expect to observe, 0.61 deaths in one corps, i.e., sometimes none, sometimes one, occasionally two, perhaps once in a while three, and very rarely more. This is the classic Poisson situation: a rare event, whose average rate is small, with observations made over many small intervals of time.
Poisson distribution The distribution of the count X of successes in the Poisson setting is the Poisson distribution with mean μ . The parameter μ is the mean number of successes per unit of measure. The possible values of X are the whole numbers 0, 1, 2, 3, …. If k is any whole number 0 or greater, then P( X = k ) = ( e - μ μ k )/ k ! The standard deviation of the distribution is the square root of μ .

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