Download presentation

Presentation is loading. Please wait.

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
**All exact science is dominated**

by the idea of approximation. Bertrand Russell ( )

3
**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

4
**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} • • • • • • • • • • • • • • •

5
**Random Variables Types: Discrete: Bernoulli, Binomial, Poisson**

Continuous: Exponential, Normal

6
Random Variables Bernoulli Binomial Poisson

7
**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

8
**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)

9
**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?

10
**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

11
**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

12
**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) = p = 0.64

13
Binomial Coefficient Factorial notation: So,

14
**Binomial Coefficient:**

By convention: 0! = 1 Binomial Coefficient:

15
**Binomial Distribution**

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

16
**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) = p = 0.64

17
**Binomial Distribution**

Binomial with n=10 and p=0.5

18
**Binomial Distribution**

Binomial with n=10 and p=0.29

19
**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)

20
**Binomial Distribution - Moments**

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

21
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 .

22
Poisson Distribution Source:

23
**Poisson Distribution Mean = For the Poisson one parameter: np**

Binomial Mean = Variance = np np(1-p) np when p is small

24
**Poisson Distribution - Example**

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

25
Poisson Distribution Poisson with =2.4

Similar presentations

Presentation is loading. Please wait....

OK

Random Variables Lecture Lecturer : FATEN AL-HUSSAIN.

Random Variables Lecture Lecturer : FATEN AL-HUSSAIN.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on effect of global warming on weather today Ppt on important places in india Ppt on games and sports in india Ppt on acids bases and salts for class 7 Ppt on special types of chromosomes mutation Ppt on artificial intelligence and robotics Ppt on types of clothes Ppt on formal letter writing Ppt on life study of mathematician Ppt on good leadership