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

All exact science is dominated by the idea of approximation. Bertrand Russell (1872-1970)

Random Variables 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 • • • • • • {1,2,3,4,5,6} • • • • • • • • • • • • • • 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} • • • • • • • • • • • • • • •

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

Random Variables Bernoulli Binomial Poisson

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

Binomial Distribution 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)

Can the binomial distribution be used in Binomial - Examples Can the binomial distribution be used in the settings below? 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?

Binomial - Example 1 Trial P(0) = 1-p = 0.2 P(1) = p = 0.8 e.g. Binomial - Example 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

Binomial - Example 2 Trials: P(0 vaccinated) = (1-p)2 e.g. 2 trials Binomial - Example 2 Trials: Trials Probability #succ. Prob (0,0) (1-p) 0.04 (0,1) p 1 0.16 (1,0) (1,1) 2 0.64 P(0 vaccinated) = (1-p)2 P(1 vaccinated) = 2p(1-p) P(2 vaccinated) = p2

Binomial - Example X  number of successes e.g. continued Binomial - Example Experiment: Sample two villagers at random and determine whether they are vaccinated 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) = p2 = 0.64

Binomial Coefficient Factorial notation: So,

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

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

N=2 trials; X=num. successes Binomial Distribution 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) = p2 = 0.64

Binomial Distribution Binomial with n=10 and p=0.5

Binomial Distribution Binomial with n=10 and p=0.29

Binomial Distribution - Moments Mean and variance Binomial Distribution - Moments 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)

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

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

Poisson Distribution Source: http://en.wikipedia.org/wiki/Poisson_distribution

Poisson Distribution Mean =  For the Poisson one parameter:  np Binomial Mean =  Variance =  np np(1-p)  np when p is small

Poisson Distribution - Example e.g. Probability of an accident in a year is 0.00024. So in a town of 10,000, the rate = np = 10,000 x 0.00024 = 2.4

Poisson Distribution Poisson with  =2.4