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Unformatted text preview: 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 X...
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This note was uploaded on 05/12/2010 for the course APPLIED ST 2010 taught by Professor Various during the Spring '10 term at Universidad Nacional Agraria La Molina.
- Spring '10