4. Discrete Probability Distributions. Introduction & Binomial Distribution

Transcription

1 4. Discrete Probability Distributions Introduction & Binomial Distribution

2 Aim & Objectives 1

3 Aims u Introduce discrete probability distributions v Binomial distribution v Poisson distribution 2

4 Objectives u Define random and Bernoulli variables u Describe a probability distribution u Compute the mean and variance of discrete random variables u Describe and utilise the Binomial distribution u Describe and utilise a Poisson distribution 3

5 4.1 Motivating Exercises 4

6 Motivating Exercise 1: Summer 2011 Q3b Scenario The probability of being in blood group B is 9%. You randomly select 6 people. Questions to Explore (i) How likely is it that none of them are blood group B? [11 marks] (ii) How likely is it that at most two are blood group B? [8 marks] (iii) What is the mean number in blood group B? [3 marks] (iv) What is the standard deviation of the number in blood group B? [3 marks] 5

7 Motivating Exercise 2: Summer 2011 Q4b Scenario The number of tadpoles scattered randomly through a pond follows a Poisson distribution with mean of 15 per litre. Questions to Explore I. Find the probability that a random sample of 1 litre of pond water will contain 5 tadpoles. [4 marks] II. Find the probability that a random sample of 0.1 litres of pond water will contain more than one tadpole. [8 marks] III. What is the probability that there will be exactly 2 tadpoles in a random sample of 0.1 litres of pond water? [6 marks] IV. What is the probability that there will be exactly 20 tadpoles in 2 litres of pond water? [6 marks] V. What is the mean number of tadpoles in a random sample of 1 litre of pond water? [2 marks] VI. What is the standard deviation of the number of tadpoles in a random sample of 1 litre of pond water? [2 marks] 6

8 4.2 Introduction 7

9 Random Variable u A random variable, X, is the numerical outcome of an experiment u Examples Experiment: Roll of a dice RV: value of the face (1, 2, 3, 4, 5, 6) Experiment: Flip of a coin: RV: 1 = Head, 2 = Tail Experiment: Opinion poll RV: 1 if in favour, 0 if not 8

10 Random Variable u For some experiments the outcomes are not numeric u A coding system is used to assign numbers to the outcomes u A random variable, X, is discrete if it can only have integer values u Examples of discrete RV s are counts of items v number of disease outbreaks per year v number of glassware breakages per week v number of hours per week spent on exercise v number of minor oil leakages in a region per month u It is not possible for the variable to have values that are fractions or decimals 9

11 Bernoulli Variable u Bernoulli variable is a very common and simple type of discrete RV u A Bernoulli variable has only two possible values: v 0 and 1 u Variable is appropriate for experiments in which there are only two outcomes u Example Experiment: Determination of whether a disease is present Possible Outcomes: {disease, no disease} Code 1= disease, 0 = no disease The outcome is a Bernoulli variable 10

12 Probability Distribution u Associated with each value (outcome) of a discrete RV is its probability of occurrence u Probability distribution is the way in which the total probability of 1 is distributed among the possible values of the discrete RV u Probability distribution of a discrete RV, is defined for every possible value, x, as P(X = x) 11

13 Probability Distribution u Probability distributions can be estimated by sampling from the population u A frequency distribution is calculated from the sample u Frequency in each class or each class interval is converted to a relative frequency u Relative frequencies are the estimated probabilities 12

14 Probability Distribution: Example u Suppose we wish to know that the probability distribution of eye colour of students Colour Number of Students Relative Frequency Estimated Probability Blue Brown Other u As we will see later, in certain circumstances, the probability distribution may also be generated by a mathematical function 13

15 4.3 Mean and Variance of Discrete Random Variable 14

16 Exercise u Consider a random variable, X, with the following probability distribution: X Probability K 15

17 Exercise X Probability K u What is the value of K? Why? 0.1, probabilities sum to zero 16

18 Exercise u What is the expected value of X? X Probability X x Probability Sum u Expected value of X = Mean value of X= E(X) =

19 Exercise u What is the variability (standard deviation)? u E(X) = 0.65 X Probability X 2 X 2 x Probability Sum u V(X)=E(X 2 )-[E(X)] 2 = = u SD(X) = =1.01 E(X 2 ) 18

20 Formulae u Mean of a discrete RV, X, is given by: E(X) = Σ {x} xp(x) u Variance of a discrete RV, X, is given by: V(X) = E(X 2 ) [E(X)] 2 E(X)= xp(x) { x} 2 2 E(X )= x P(x) { x} 19

21 4.4 Binomial Distribution 20

22 Binomial Distribution u Binomial experiment has the following properties: v a sequence of n Bernoulli trials v each trial is independent v the probability of success in a single trial is constant u Random variable X, defined to be the number of successes in the n trials, is a Binomial variable u Binomial variable has two parameters: v n: the number of trials v p: the probability of success in a single trial. u If a RV X, is Binomial, we denote this by: X ~ Bin(n, p) 21

23 Binomial Distribution u Formula for generating the probability distribution for a Binomial variable is given by: v where r can be any value from 0 to n u Formula for calculating n r n-r P(X=r)=Crp (1-p) v Known as a combination and is a subset of r items taken from a total of n items C = n r n! r!(n-r)! 22

24 Binomial Distribution u For example: 20 20! 20x19 C 2 = = = 190 2!(20-2)! 2x1 u Most calculators have this function 23

25 Binomial Distribution Mean and Variance of a Binomial Variable 24

26 Mean and Variance of a Binomial Variable If X ~ Bin(n, p), then E(X) = np V(X) = np(1 p). 25

27 Binomial Distribution: In-class Exercise Summer 2010 Q3(b) A population of paradise flycatchers has 60% brown males and 40% white. Your field assistant captures 5 male flycatchers at random. u How likely is it that none of the flycatchers are brown? [11 marks] u How likely is it that at most one flycatcher is brown? [8 marks] u What is the mean number of brown flycatchers? [3 marks] u What is the standard deviation of the number of brown flycatchers? [3 marks] 26

28 Binomial Distribution: In-class Exercise Summer 2009 Q2(b) In a certain developing country, 30% of children are malnourished. If a random sample of 5 children from this country is chosen how likely is it that (i) None of the children are malnourished? (ii) At least 1 child is malnourished? What is the mean number of children that are malnourished? What is the standard deviation of the number malnourished? 27

29 Summary u Defined v Random variable v Bernoulli variable u Introduced probability distribution u Computed the mean and variance of a discrete RV u Described the Binomial distribution u Applied the Binomial distribution 28

30 What next? u Discrete probability distributions v Poisson distribution 29

31 4. Discrete Probability Distributions Poisson Distribution

32 Summary of Previous Lecture u A Bernoulli variable has only two possible values: 0 and 1 u Probability distribution is the way in which the total probability of 1 is distributed among the possible values of the discrete RV u Binomial experiment has a sequence of n Bernoulli trials, each trial is independent, the probability of success in a single trial is constant u Binomial variable has two parameters, n: the number of trials, p: the probability of success in a single trial u If a RV X, is Binomial, we denote this by: X ~ Bin(n, p) P(X=r)=C p (1-p) n r n-r r u Mean of a Binomial RV= E(X)=np; Variance of a Binomial RV=V(X)=np(1-p) 31

33 Aim & Objectives 32

34 Objectives u Define random and Bernoulli variables u Describe a probability distribution u Compute the mean and variance of discrete random variables u Describe and utilise the Binomial distribution u Describe and utilise a Poisson distribution 33

35 4.5 Poisson Distribution 34

36 Poisson Distribution u A Poisson variable is typically a count of relatively rare occurrences Examples u Number of outbreaks of disease per year u Number of barnacles per m 2 of rock u Number of sightings of flocks of migratory birds per day 35

37 Poisson Distribution: Example u The Poisson distribution has one parameter, λ, the mean number of occurrences per unit measurement u If a RV X, is Poisson, we denote this by: X ~ Poi(λ) u The formula for generating the probability distribution for a Poisson variable is given by: P(X=r) = -λ r e λ r! v where r can be any value from 0 upwards 36

38 Poisson Distribution Mean and Variance of a Poisson Variable

39 Mean and Variance of a Poisson Variable If X ~ Poi(λ), then E(X) = λ V(X) = λ. 38

40 Poisson Distribution: Example u Since 1994, the number of outbreaks of Ebola in Gabon is a Poisson variable with mean 0.7 outbreaks per year. In the next year what is the probability of I. no outbreaks? II. III. at most 2 outbreaks? at least 2 outbreaks? 39

41 Poisson Distribution: In-class Exercise Summer 2010 Q4(b) An experiment measures the number of particle emissions from a radioactive substance. The number of emissions follows a Poisson distribution with mean number of emissions per week of 0.3. I. Find the probability that in any given week there will be at least one emission. [8 marks] II. III. IV. Find the probability that in any given week there will be exactly two emissions. [4 marks] What is the probability that there will be exactly 2 emissions in the next fortnight? [6 marks] What is the probability that there will be exactly 2 emissions in the next year? [6 marks] V. What is the mean number of particle emissions in one week? [2 marks] VI. What is the standard deviation of the number of particle emissions in one week? [2 marks] 40

42 Poisson Distribution: In-class Exercise Summer 2009 Q2(c) The mean number of accidents per year in a large factory is five. Assuming the number of accidents follows a Poisson distribution find the probability that in the current year there will be: I. Three accidents II. At most one accident What is the probability that there will be exactly 10 accidents in the next 2 years? 41

43 Summary u Described the Poisson distribution u Applied the Poisson distribution 42

44 What next? u Motivating exercises 43

45 Summary of Poisson Distribution u A Poisson variable is typically a count of relatively rare occurrences u The Poisson distribution has one parameter, λ, the mean number of occurrences per unit measurement u If a RV X, is Poisson, we denote this by: X ~ Poi(λ) P(X=r) = e λ r! -λ r u Mean of a Poisson RV= E(X)= λ; Variance of a Poisson RV=V(X)= λ 44

46 4.1 Motivating Exercises 45

47 Motivating Exercise 1: Summer 2011 Q3b Scenario The probability of being in blood group B is 9%. You randomly select 6 people. Questions to Explore (i) How likely is it that none of them are blood group B? [11 marks] (ii) How likely is it that at most two are blood group B? [8 marks] (iii) What is the mean number in blood group B? [3 marks] (iv) What is the standard deviation of the number in blood group B? [3 marks] 46

48 Motivating Exercise 2: Summer 2011 Q4b Scenario The number of tadpoles scattered randomly through a pond follows a Poisson distribution with mean of 15 per litre. Questions to Explore I. Find the probability that a random sample of 1 litre of pond water will contain 5 tadpoles. [4 marks] II. Find the probability that a random sample of 0.1 litres of pond water will contain more than one tadpole. [8 marks] III. What is the probability that there will be exactly 2 tadpoles in a random sample of 0.1 litres of pond water? [6 marks] IV. What is the probability that there will be exactly 20 tadpoles in 2 litres of pond water? [6 marks] V. What is the mean number of tadpoles in a random sample of 1 litre of pond water? [2 marks] VI. What is the standard deviation of the number of tadpoles in a random sample of 1 litre of pond water? [2 marks] 47