Probability Theory and Simulation Methods. April 6th, Lecture 19: Special distributions

Transcription

1 April 6th, 2018 Lecture 19: Special distributions

2 Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter 5, 7: Special distributions Week 10 Chapters 8, 9, 10: Bivariate and multivariate distributions Week 12 Chapter 11: Limit theorems

3 Chapter 5: Special discrete distributions Bernoulli random variables Binomial random variables Poisson random variables Other discrete random variables Geometric random variables Negative binomial random variables Hypergeometric random variables

4 Bernoulli random variables Sample space: {s, f} The random variable defined by X(s) = 1 and X(f) = 0 is called a Bernoulli random variable The pmf of a Bernoulli random variable is p if x = 1 p(x) = 1 p if x = 0 0 elsewhere where p is a parameter, referred to as the probability of a success E(X) = p and Var(X) = p(1 p)

5 Binomial random variables Definition If n Bernoulli trials all with probability of success p are performed independently, then X, the number of successes is called a binomial random variable with parameters n and p.

6 Binomial random variables: example Example A restaurant serves 8 entrees of fish, 12 of beef, and 10 of poultry. If customers select from these entrees randomly, what is the probability that two of the next four customers order fish entrees?

7 Poisson random variables Binomial random variables B(n, p) are useful, but their probabilities are horrible ( ) n P(X = i) = p i (1 p) n i i computationally difficult, especially when n is large Simeon-Denis Poisson (1837): Assume that n is large, p < 0.1 and λ = np < 10, then when n approaches infinity Law of rare events P(X = i) e λ λ i i!

8 Poisson random variables also used to model the number of success, but only when n is large and the average number of success is moderate (< 10)

9 pmf of Poisson random variables

10 Poisson random variables

11 Poisson distribution in reality The Poisson distribution also appears in connection with various phenomena of discrete properties whenever the probability of the phenomenon happening is constant in time or space. the number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry the number of yeast cells used when brewing Guinness beer the number of phone calls arriving at a call centre within a minute the number of deaths per year in a given age group. the number of jumps in a stock price in a given time interval.

12 Poisson process

13 Poisson process

14 Other discrete random variables

15 Geometric random variables Consider an experiment in which independent Bernoulli trials (with parameter p) are performed until the first success occurs. The sample space for such an experiment is S = {s, fs, ffs, fffs,..., ff... fs,...}. Let X be the number of experiments until the first success occurs. Then X is a discrete random variable called geometric.

16 Geometric random variables Problem 1 Prove that p(x) is a probability mass function 2 Compute E(X)

17 Geometric random variables Example From an ordinary deck of 52 cards we draw cards at random, with replacement, and successively until an ace is drawn. What is the probability that exactly 10 draws are needed?