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1 Introduction to Biostatistics (PUBHLTH 540) Lecture 7: Binomial and Poisson Distributions Acknowledgement: Thanks to Professor Pagano (Harvard School of Public Health) for lecture material

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2 All exact science is dominated by the idea of approximation. Bertrand Russell ( )

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3 Variable: measurable characteristic Random Variable: variable that can have different outcomes of an experiment, determined by chance Examples: X = outcome of roll of a die, Y = outcome of a coin toss, Z = height Random Variables

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4 Random Variable is a function that assigns specific numerical values to all possible outcomes of experiment Probability distributions are associated with random variables to describe the probabilities of the various outcomes of an experiment {1,2,3,4,5,6}

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5 Random Variables Types: –Discrete: Bernoulli, Binomial, Poisson –Continuous: Exponential, Normal

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6 Random Variables Bernoulli Binomial Poisson

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7 Dichotomous (Bernoulli): X = 0 or 1 P(X=1) = p P(X=0) = 1-p e.g. Heads, Tails True, False Success, Failure Random Variables - Bernoulli When outcomes of experiment are binary

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8 A sequence of independent Bernoulli trials (n) with constant probability of success at each trial (p) and we are interested in the total number of successes (x). Assumptions: N trials of an experiment Each experiment results in one of 2 outcomes (binary) Each trial is independent of the other trials In each trial, the probability of ‘success’ is constant (p) Binomial Distribution

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9 Binomial - Examples 10 tosses of a coin – Yes/No? 10 rolls of a die – Yes/No? 10 rolls of a die and the number time it turns up a 6 – Yes/No? Number of individuals who have a particular disease in a town – Yes/No? Can the binomial distribution be used in the settings below?

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10 Suppose that 80% of the villagers should be vaccinated. What is the probability that at random you choose a vaccinated villager? 1 success (vaccinated person) 0 failure (unvaccinated person) 1 Trial P(0) = 1-p = 0.2 P(1) = p = 0.8 e.g. Binomial - Example

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11 2 Trials: TrialsProbability#succ.Prob (0,0)(1-p) (0,1)(1-p)p10.16 (1,0)p(1-p)10.16 (1,1)pp20.64 P(0 vaccinated) = (1-p) 2 P(1 vaccinated) = 2p(1-p) P(2 vaccinated) = p 2 e.g. 2 trials Binomial - Example

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12 X number of successes n = 2, the number of trials P(X=0) = (1-p) 2 = 0.04 P(X=1) = 2p(1-p) = 0.32 P(X=2) = p 2 = 0.64 e.g. continued Binomial - Example Experiment: Sample two villagers at random and determine whether they are vaccinated

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13 So, Factorial notation: Binomial Coefficient

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14 By convention: 0! = 1 Binomial Coefficient: Binomial Coefficient

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15 X = number of successes in n trials Parameters: p = probability of success n = number of trials Binomial Distribution

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16 N=2 trials; X=num. successes P(X=0) = (1-p) 2 = 0.04 P(X=1) = 2p(1-p) = 0.32 P(X=2) = p 2 = 0.64 Binomial Distribution

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17 Binomial with n=10 and p=0.5 Binomial Distribution

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18 Binomial with n=10 and p=0.29 Binomial Distribution

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19 For X ~ Binomial(n,p) (i.e. n = Num. Trials, p = Probability of success in each trial) Then Mean = E(X) = np Variance = Var(X) = np(1-p) Mean and variance Binomial Distribution - Moments

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20 e.g. p=0.5 n=10 Mean = np = = 5 Variance =np(1-p) = 10(0.5)(0.5) =2.5 Binomial Distribution - Moments

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21 1.The probability an event occurs in the interval is proportional to the length of the interval. 2.An infinite number of occurrences are possible. 3.Events occur independently at a rate. Poisson Distribution X=number of occurrences of event in a given time period

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22 Poisson Distribution Source:

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23 For the Poisson one parameter: Mean = Variance = np np(1-p) np when p is small Poisson Distribution PoissonBinomial

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24 l= np = 10,000 x = 2.4 e.g. Probability of an accident in a year is So in a town of 10,000, the rate Poisson Distribution - Example

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25 Poisson with =2.4 Poisson Distribution

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