Probability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions

Transcription

1 April 9th, 2018 Lecture 20: 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 Special discrete distributions

4 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

5 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)

6 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.

7 Poisson random variables

8 pmf of Poisson random variables

9 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.

10 Geometric random variables

11 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?

12 Negative binomial random variables Consider an experiment in which independent Bernoulli trials (with parameter p) are performed until the r th success occurs. Let X be the number of experiments until the r th success occurs. Then X is a negative binomial random variables with parameters (r, p). A negative binomial random variables with parameters (1, p) is a geometric random variable

13 Negative binomial random variables

14 Hypergeometric random variables Suppose that, from a box containing D defective and ND non-defective items, n are drawn at random and without replacement. Let X be the number of defective items drawn. Then X is a discrete random variable with the set of possible values {0, 1,...n} and probability mass function

15 Hypergeometric random variables

16 Special continuous distributions

17 Chapter 5: Special continuous distributions Uniform random variables Normal random variables Exponential random variables Other discrete random variables Gamma distribution Beta distributions

18 Uniform random variables Suppose that X is the value of the random point selected from an interval (a, b) such that every point is equally likely to be chosen Then X is called a uniform random variable over (a, b)

19 Uniform random variables Problem The pdf of X has the form c if a < x < b f(x) = 0 elsewhere where c is some unknown constant. 1 Compute c 2 Write down the formula for the distribution function of X 3 Compute E(X) and Var(X)

20 Uniform random variables

21 Uniform random variables

22 Uniform random variables Problem A person arrives at a bus station every day at 7:00 A.M. If a bus arrives at a random time between 7:00 A.M. and 7:30 A.M., what is the average time spent waiting?

23 Normal random variables

24 N(µ, σ 2 ) E(X) = µ, Var(X) = σ 2

25 Standard normal distribution If Z is a normal random variable with parameters µ = 0 and σ = 1, then the pdf of Z is f(z) = 1 2π e z2 2 and Z is called the standard normal distribution E(Z) = 0, Var(Z) = 1 The cumulative distribution function of the standard normal distribution is: Φ(z) = P(Z z) = z f(y) dy

26 Φ(z) Φ(t) = P(Z t) = z f(y) dy

27 Φ(z)

28 Shifting and scaling normal random variables Problem Let X be a normal random variable with mean µ and standard deviation σ. Then Z = X µ σ follows the standard normal distribution.

29 Shifting and scaling normal random variables