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# binomsum - Statistics binomial distribution and proportions...

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Statistics: binomial distribution and proportions The binomial distribution The binomial scenario : the binomial distribution is an appropriate tool in the following circum- stances: 1. There are n “trials”. 2. The trials are independent. 3. On each trial, only two things can happen, “success”and “failure”. 4. The probability of a “success” is the same on each trial. It is usual to call this probability p . 5. We count the total number of “successes”. The total number of successes that we see in such a scenario is a discrete random variable, X say, that can take any integer value between 0 and n inclusive, e.g. we see no heads, 1 head, . . . , all heads. The random variable is said to have a binomial distribution with parameters n, p , abbreviated X b ( n, p ). It is easy to show that if X b ( n, p ) then its probability distribution is P ( X = k ) = n k p k (1 - p ) n - k = n ! ( n - k )! k ! p k (1 - p ) n - k , k = 0 , 1 , . . . , n. We can show that the probabilities in this distribution add up to one quite easily. You might remember that the binomial theorem states that, for any two numbers a and b , ( a + b ) n = n k =0 n k a n b n - k .

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binomsum - Statistics binomial distribution and proportions...

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