Chapters 3.2 Discrete distributions

Transcription

1 Chapters 3.2 Discrete distributions In this section we study several discrete distributions and their properties. Here are a few, classified by their support S X. There are of course many, many more. For each of these, we will discuss the PMF, moments, parameters, relationship with other distributions, and potential applications. S X Families {1, 2,..., N} Uniform; Hypergeometric {0, 1} Bernoulli {0, 1,..., n} Binomial {0, 1,...} Poisson; Negative binomial; Geometric ST521 Chapter 3.2 Page 1

2 Discrete uniform distribution If X Uniform(N) then P (X = x) = 1/N for x S X = {1,..., N}. This family has a single parameter, N {1, 2,...}. E(X)= V(X)= ST521 Chapter 3.2 Page 2

3 Hypergeometric distribution Say there are N balls in a bag: M are red, N M are blue. We randomly select k balls (k {1,..., N}), and denote X as the number of red balls. X Hypergeometric(M, N, k) There are three parameters: M, N, and k. S X = f X (x) = Examples: ST521 Chapter 3.2 Page 3

4 Example: There are 5,000 Republicans and 5,000 Democrats in Wake County. We randomly select k people, and denote X as the number of Republicans in the survey. Say k = 10, what is the probability of a 50/50 split in the survey, X = 5? Say k = 10, 000, what is the probability of a 50/50 split in the survey, X = 5, 000? ST521 Chapter 3.2 Page 4

5 The mean and variance are E(X) = km N and V(X) = KM N (N M)(N k) N(N 1). ST521 Chapter 3.2 Page 5

6 Binomial distribution If X is the number of successes in n trials, conducted so that 1. the trials are independent 2. the probability of a success for each trial is p (0, 1) then X Binomial(n, p). S X = There are two parameters: n and p. If n = 1, then X Bernoulli(p). Then X {0, 1}. Real-life examples: ST521 Chapter 3.2 Page 6

7 f X (x) = ST521 Chapter 3.2 Page 7

8 The mean and variance are: ST521 Chapter 3.2 Page 8

9 Example: Over many years we know that if patients are given the standard care, p = 0.1 is the probability of the recurrence of the disease. We give a new drug to 100 patients. To test whether the new drug is more effective than the standard care, we decide to conclude that the new drug is better if X, the number of patients on the new drug with a recurrence, is less than t = 5. Say the drug doesn t work, and p = 0.1 for the new drug as well. What is the probability we incorrectly conclude the new drug works? Statistical tests are derived by picking t so that this probability is α = 0.05 (Chapter 8). ST521 Chapter 3.2 Page 9

10 ˆp = X/n is the sample proportion. ST521 Chapter 3.2 Page 10

11 Comparing the binomial and hypergeometric distributions Example: Wake County voting with N = 1, 000 voters, M = 500 Republicans, and p = M/N = 0.5 proportion Republicans. We take a sample of size k = 700, and observe X = 360 and ˆp = 360/700 = Two ways to compute the standard error of ˆp: 1. Sampling without replacement: 2. Sampling with replacement: ST521 Chapter 3.2 Page 11

12 Poisson distribution If events occur at a rate of λ per minute, independent of each other, and X is the number of events that occur in one minute, then X Poisson(λ). Examples: S X = {0, 1, 2,...}. P(X = x) = exp( λ)λx x!. E(X) = V(X) = ST521 Chapter 3.2 Page 12

13 Examples: Say we typically get s at a rate of two per hour, independent over time. Let X be the number of s we get in a day. If we observed X = 70 s in a day, is this evidence of unusual activity? ST521 Chapter 3.2 Page 13

14 Comparing the Poisson and binomial ST521 Chapter 3.2 Page 14

15 Continued: ST521 Chapter 3.2 Page 15

16 A more formal proof: ST521 Chapter 3.2 Page 16

17 Negative binomial distribution If trials are independent with success probability p, we continue trying until we get r successes, and denote X as the number of trials until we get r successes, then X NegBinom(r, p). Examples: S X = {r, r + 1,...}. P(X = x) = E(X) = V(X) = ST521 Chapter 3.2 Page 17

18 Alternative definition: Y is the number of failures before the r successes. This is related to the previous definition as Y = X r. S X = {0, 1,...}. P(X = x) = E(X) = V(X) = ST521 Chapter 3.2 Page 18

19 Comparing the negative binomial and Poisson distributions The negative binomial and Poisson have the same support, S X = {0, 1, 2,...}, and are both often used to model count data. For the Poisson, E(X) = V(X) = λ. What if the mean is bigger than the variance? For the negative binomial, defining E(X) = r(1 p)/p = λ, the variance is λ/p, which is larger than the mean since p (0, 1). So the negative binomial is more flexible because it has a second parameter which allows the variance to be greater than the mean, which often improves fit to data. ST521 Chapter 3.2 Page 19

20 Geometric distribution If r = 1 and X is the number of trials until the first success, then X Geometric(p). Examples: S X = {1, 2,...}. P(X = x) = E(X) = V(X) = ST521 Chapter 3.2 Page 20