# 1 - ECMT 1020 Summer School 2009 Lecture 1 Binomial Normal...

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Lecture 1 Binomial, Normal approximation to the Binomial, Proportions ECMT 1020 Summer School 2009

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Experiment: Toss 2 Coins. Let X = # heads. T T Discrete Probability Distribution 4 possible outcomes T T H H H H Probability Distribution 0 1 2 X X Value Probability 0 1/4 = 0.25 1 2/4 = 0.50 2 1/4 = 0.25 0.50 0.25 Probability
Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal Uniform Exponential Ch. 5 Ch. 6

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The Binomial Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions
Binomial Probability Distribution A fixed number of observations, n e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse Two mutually exclusive and collectively exhaustive categories e.g., head or tail in each toss of a coin; defective or not defective light bulb Generally called “success” and “failure” Probability of success is p, probability of failure is 1 – p Constant probability for each observation e.g., Probability of getting a tail is the same each time we toss the coin

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Binomial Probability Distribution (continued) Observations are independent The outcome of one observation does not affect the outcome of the other Two sampling methods Infinite population without replacement Finite population with replacement
Possible Binomial Distribution Settings A manufacturing plant labels items as either defective or acceptable A firm bidding for contracts will either get a contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it

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Rule of Combinations The number of combinations of selecting X objects out of n objects is X)! (n X! n! C x n - = where: n! =(n)(n - 1)(n - 2) . . . (2)(1) X! = (X)(X - 1)(X - 2) . . . (2)(1) 0! = 1 (by definition)
P(X) = probability of X successes in n trials, with probability of success p on each trial X = number of ‘successes’ in sample, (X = 0, 1, 2, . .., n ) n = sample size (number of trials or observations) p = probability of “success” P(X) n X ! n X p (1- p ) X n X ! ( )! = - - Example: Flip a coin four times, let x = # heads: n = 4 p = 0.5 1 - p = (1 - 0.5) = 0.5 X = 0, 1, 2, 3, 4 Binomial Distribution Formula

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Example: Calculating a Binomial Probability What is the probability of one success in five observations if the probability of success is .1? X = 1, n = 5, and p = 0.1 0.32805 .9) (5)(0.1)(0 0.1) (1 (0.1) 1)! (5 1! 5! p) (1 p X)! (n X! n! 1) P(X 4 1 5 1 X n X = = - - = - - = = - -
n = 5 p = 0.1 n = 5 p = 0.5 Mean 0 .2 .4 .6 0 1 2 3 4 5 X P(X) .2 .4 .6 0 1 2 3 4 5 X P(X) 0 Binomial Distribution The shape of the binomial distribution depends on the values of p and n Here, n = 5 and p = 0.1 Here, n = 5 and p = 0.5

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Binomial Distribution Characteristics Mean Variance and Standard Deviation
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1 - ECMT 1020 Summer School 2009 Lecture 1 Binomial Normal...

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