Presentation is loading. Please wait.

Presentation is loading. Please wait.

1 Introduction to Biostatistics (PUBHLTH 540) Lecture 7: Binomial and Poisson Distributions Acknowledgement: Thanks to Professor Pagano (Harvard School.

Similar presentations


Presentation on theme: "1 Introduction to Biostatistics (PUBHLTH 540) Lecture 7: Binomial and Poisson Distributions Acknowledgement: Thanks to Professor Pagano (Harvard School."— Presentation transcript:

1 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

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

3 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

4 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}

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

6 6 Random Variables Bernoulli Binomial Poisson

7 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

8 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

9 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?

10 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

11 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

12 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

13 13 So, Factorial notation: Binomial Coefficient

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

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

16 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

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

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

19 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

20 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

21 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

22 22 Poisson Distribution Source:

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

24 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

25 25 Poisson with =2.4 Poisson Distribution


Download ppt "1 Introduction to Biostatistics (PUBHLTH 540) Lecture 7: Binomial and Poisson Distributions Acknowledgement: Thanks to Professor Pagano (Harvard School."

Similar presentations


Ads by Google