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# Lecture5U3 - ECON1203/ECON2292 Business and Economic...

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Unformatted text preview: ECON1203/ECON2292 Business and Economic Statistics Unit 3 Lecture 5 2 Lecture 5 topics Discrete probability distributions Binomial distribution Continuous probability distributions Uniform distribution Key references Keller 7.4, 8.1 Learning Outcomes: You should be able to Find probabilities with binomial distributions Find probabilities with uniform distributions Introduction to binomial distribution Previously derived probability distributions from first principles Recall distribution of males in gender composition example But probability distribution of heads in 3 coin tosses has exactly the same distribution Context of examples is different But each of (three) ‘trials’ has one of two possible outcomes (male/female, or head/tail) Generic distribution here is the binomial 3 Introduction to binomial distribution… Based on notion of a binomial experiment consisting of a sequence of trials Assumptions for binomial experiment Sequence of fixed number of n trials Each trial has 2 outcomes, arbitrarily denoted success & failure Fixed probability of success (failure) p (1- p ) over trials Trials are independent Under these assumptions have a sequence of Bernoulli trials 4 Binomial random variable Have a sequence of Bernoulli rv’s X 1 , X 2 , …, X n where X i = 1 if success & X i = 0 for a failure Under assumptions made this is a sequence of independent & identically distributed (iid) rv’s E.g. If n = 5 , one possible outcome of the sequence of trials is 0, 1, 1, 0, 1 Another is 1, 1, 0, 0, 0 etc. 5 Binomial random variable… Often interested in rv’s constructed from other rv’s Consider rv formed by summing these n Bernoulli rv’s X = X 1 + X 2 + … + X n X represents number of success in n trials & is called a binomial random variable Characterized by 2 parameters n & p Once we know the parameter values we know everything about the rv & its probability distribution 6 Binomial distribution In gender composition example can calculate P (MMF) = P ( X 1 =1, X 2 =1, X 3 =0)= pp (1- p ) = p 2 (1- p ) but there are 3 ways to obtain 2 Males (others being MFM or FMM) so P (2 Males) = P ( X = 2) = 3 p 2 (1- p ) Coefficient of 3 on the probability is exactly the total number of combinations when choosing 2 from 3 7 Binomial distribution… 3 )! 2 3 ( ! 2 ! 3 then 2 & 3 If 1 0! and ) 1 )( 2 )...( 2 )( 1 ( ! where )! ( ! ! from choose to many ways how i.e. formula ial combinator the need Here ns) permutatio and ons (combinati techniques counting Recall 3 2 C x n n n n n x n x n C n x n x 8 9 Binomial distribution… n x p p x n x n x P x X P p n x x n x , , 1 , for ) 1 ( )! ( !...
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Lecture5U3 - ECON1203/ECON2292 Business and Economic...

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