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# Binomial Distributions STAT 1010

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This 2 page Class Notes was uploaded by Susannah Gilmore on Wednesday March 23, 2016. The Class Notes belongs to STAT 1010 at University of Virginia taught by in Spring 2016. Since its upload, it has received 10 views. For similar materials see Introduction to Statistics in Statistics at University of Virginia.

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Date Created: 03/23/16

Binomial Distributions Binomial Rules In order for an experiment to be binomial, it must satisfy the following rules: 1 There are a fixed number of observations/trials, which is represented by n 2 The n observations are all independent of each other, i.e. one does not affect the other 3 Each observation falls into one of just two categories, either success or failure 4 The probability of a success, p, is the same for each observation Binomial Random variable, X: counts the number of successes in trials (n = number of trials) of a binomial experiment. (So X = number of successes) Properties of X: o X is a discrete random variable o The possible values of X are 0 - n o The binomial random variable X follows a binomial distribution Binomial Parameters 1 n: the number of observations/trials 2 p: the probability of success for any observation The Binomial Coefficient (n k) = means the number of ways the observations can be arranged The "5!" would mean to multiply 5 * 4 * 3 * 2* 1 *** Remember that a factorial of 0! Is always equal to 1 The Binomial Probability So essentiallx the n-xomial probability is the binomial coefficient multiplied by p (1-p) When solving problems, you need to first define: s - the max number that you want for x. If the probability has a greater than symbol, then you must take the complement, because the formula and excel only work for what is to the left of the x. o For example: P(x>2) has a complement of P(x≤2) p (probability of success) n (number of trials) k (how many successes is the problem asking for) Probability examples: P(x<3) means P(x=0) + P(x=1) + P(x=2) P(x≤3) means P(x=0) + P(x=1) + P(x=2) + P(x=3) Using Excel for the binomial distribution Formula =BINOM.DIST(k,n,p,c) K = the max number that you want for x. If the probability has a greater than symbol, then you must take the complement, because the formula and excel only work for what is to the left of the x. o For example: P(x>2) has a complement of P(x≤2) o if you want the probability up to 3, then it would be 3 N = the number of trials P = probability of success C can be either o 0 (not cumulative, ex: the probability of exactly 3) o 1 (if you want it cumulative, ex: the probability up to 3) **don’t forget to take the complement and subtract from 1 IF the probability is asking for a greater than symbol, because excel only takes probabilities from the left** Mean and Variance of Binomial Random Variables Mean = np (number of trials x probability) Variance = np(1-p) Standard Deviation = sqrt(varience) Normal Approximation to the Binomial When n is large, the distribution of X is approximately normal We can use the normal approximation when o np ≥ 10 and o n(1-p) ≥ 10 If these two requirements are satisfied, you can use the normal approximation (standardize to z-score, use "norm.dist" function in excel) If these requirements are not satisfied, you have to use the binomial probability formula or the "binom.dist" function in excel(above)

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