What s for today. More on Binomial distribution Poisson distribution. c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

Transcription

1 What s for today More on Binomial distribution Poisson distribution c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

2 Review: Binomial distribution Question: among the following, what are the requirements of Binomial experiment? 1 The experiment consists of a sequence of n trials 2 Each trial results in either success (S) or failure (F) 3 The trials are independent 4 The probability of success is consistent from trial to trial 5 The probability of success is always unknown c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

3 Review: Binomial distribution Question: Let X be the number of success out of 10 trials of binomial experiment with success probability 0.3 (i.e. X is a binomial random variable, Bin(10, 0.3)) 1 What is p X (x)? 2 What is E(X)? 3 What is V (X)? c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

4 What are these? The number of cars that pass through a certain point on a road during a given period of time. The number of spelling mistakes a secretary makes while typing a single page. The number of phone calls at a call center per minute. The number of times a web server is accessed per minute. The number of road kill found per unit length of road. c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

5 What are these? The number of mutations in a given stretch of DNA after a certain amount of radiation. The number of pine trees per unit area of mixed forest. The number of stars in a given volume of space. The number of light bulbs that burn out in a certain amount of time Q. What are the common characteristics among the above items? c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

6 Poisson distribution Let X be a number of events occurring in a fixed period of time if these events occur with an average rate, and are independent of the time since the last event The distribution of X is called a Poisson distribution The pmf of X depends on a parameter (let s call it λ) which gives the average rate p X (x) = p(x; λ) = e λ λ x, x = 0, 1, 2, x! c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

7 Prussian cavalryman data Using army records, von Bortkiewicz (1898) noted the chance of a Prussian cavalryman being killed by the kick of a horse The records of ten army corps were examined over 20 years, giving a total of 200 observations of one corps for a one year period The total deaths from horse kicks were 122, and the average deaths per year per corps was thus 122/200 = 0.61 c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

8 Prussian cavalryman data Let X: number of deaths occurred from horse kick in one corps in a given year Then, X p(x; λ) What is the value of λ? From our data, we know the average deaths per year per corps is 122/200 = 0.61 This number, 0.61, is λ to substitute in the Poisson formula, p X c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

9 Prussian cavalryman data What is the probability that no death occurred from horse kick in one corps in a given year? p X (0) = p(0; λ = 0.61) = e = ! What is the probability that 1 death occurred from horse kick in one corps in a given year? p X (1) = p(1; λ = 0.61) = e = ! c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

10 Prussian cavalryman data Given that p X (0) = , then over the 200 years observed, how many years with no death should we expect to find? 200 p X (0) = (or 110) years Given that p X (1) = , then over the 200 years observed, how many years with one death should we expect to find? 200 p X (1) = (or 66) years c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

11 Prussian cavalryman data For the entire set of Prussian data, let p be the pmf of the Poisson distribution (frequency for a given number of deaths per year) E be the corresponding number of years in which that number of deaths is expected to occur in our 200 samples (that is, our p value times 200) A be the actual number of years in which that many deaths were observed (the data) c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

12 Prussian cavalryman data Deaths p E A Q. How did we get all the numbers in the above table? Q. Why the values under E and A are different? What are they? c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

13 More Poisson problems It has been observed that the average number of traffic accidents on the Hollywood Freeway between 7 and 8 AM on Tuesday mornings is 1 per hour. What is the chance that there will be 2 accidents on the Freeway, on some specified Tuesday morning (per hour)? How did I get this answer? c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

14 More Poisson problems Coliform bacteria are randomly distributed in a certain Arizona river at an average concentration of 1 per 20cc of water. If we draw from the river a test tube containing 10cc of water, what is the chance that the sample contains exactly 2 coliform bacteria? How did I get this answer? c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

15 More on Poisson distribution X p(x; λ) E(X) = λ V (X) = λ c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16

16 what we will do next time This is the end of Topic 3. Next class we will learn about Topic 4 continuous distribution. c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16