1. Probability Theory and Specific Distributions (Moore Ch5 and Guan Ch6)

2. Research questions  Waiting Bus  On averagely, how many buses will arrive in a hour?  Or, we may ask how long do we need to wait at the bus stop?

3. Objectives  Bernoulli  The binomial setting  Binomial probabilities  Hypergeometric distribution  The Poisson model  Exponential distribution  The Normal approximation  Chi-square distribution  t distribution  F distribution Discrete distribution Continuous distribution

4. Bernoulli experiment  Bernoulli experiment( 白奴里試驗 ) ： An experiment with binary outcome.  Flip a coin with half-half chance  The probability function of a distribution following Bernoulli experiment is A coin is flipped 1 time. The variable X is the outcome of either head or tail. If the coin is a fair one, then

5. Binomial setting Binomial distributions ( 二項分配 ) are models for some categorical variables, typically representing the number of successes in a series of n trials. The observations must meet these requirements:  The total number of observations n is fixed in advance.  The outcomes of all n observations are statistically independent.  Each observation falls into just one of two categories: success and failure.  All n observations have the same probability of “success,” p. We record the next 50 vehicles sold at a dealership. Each buyer either purchases or leases; each vehicle sold is either an SUV or not.

6. The distribution of the count X of successes in the binomial setting is the binomial distribution with parameters n and p: B(n,p).  The parameter n is the total number of observations.  The parameter p is the probability of success on each observation.  The count of successes X can be any whole number between 0 and n. A coin is flipped 10 times. Each outcome is either a head or a tail. The variable X is the number of heads among those 10 flips, our count of “successes.” On each flip, the probability of success, “head,” is 0.5. The number X of heads among 10 flips has the binomial distribution B(n = 10, p = 0.5). Binomial distribution

7. Applications for binomial distributions Binomial distributions describe the possible number of times that a particular event will occur in a sequence of observations. They are used when we want to know about the occurrence of an event, not its magnitude.  In a clinical trial, a patient’s condition may improve or not. We study the number of patients who improved, not how much better they feel.  Was a sales transaction considered pleasant? The binomial distribution describes the number of pleasant transactions, not how pleasant they are.  In quality control we assess the number of defective items in a lot of goods, irrespective of the type of defect.

8. Binomial probabilities The number of ways of arranging k successes in a series of n observations (with constant probability p of success) is the number of possible combinations (unordered sequences). This can be calculated with the binomial coefficient: If X obeys binomial distribution with parameters n and p, we usually note as X~B(n,p).

9. Binomial formulas  The binomial coefficient “n_choose_k” uses the factorial notation “!”.  The factorial n! for any strictly positive whole number n is: n! = n × (n − 1) × (n − 2) × · · · × 3 × 2 × 1  For example: 5! = 5 × 4 × 3 × 2 × 1 = 120  Note that 0! = 1.

10. Finding binomial probabilities: tables  You can also look up the probabilities for some values of n and p in Table C in the back of the book.  The entries in the table are the probabilities P(X = k) of individual outcomes.  The values of p that appear in Table C are all 0.5 or smaller. When the probability of a success is greater than 0.5, restate the problem in terms of the number of failures.

11. Customer satisfaction Each consumer has probability of 0.25 of preferring your product over a competitor’s product. If we question five consumers, what is the probability that exactly three of them prefer your product?  Use Excel “=BINOMDIST(number_s,trials,probability_s,cumulative)” P(x = 3) = BINOMDIST(3, 5, 0.25, 0) = 0.08789  P(x = 3) = (n! / a!(n  a)!)p a (1  p) n-a = (5! / ((3!)(2!)) * 0.25 3 * 0.75 2 P(x = 3) = (5*4) / (2*1) * 0.25 3 * 0.75 2 P(x = 3) = 10 * 0.015625 * 0.5625 = 0.08789

12. Binomial mean and standard deviation The center and spread of the binomial distribution for a count X are defined by the mean  and standard deviation  : Effect of changing p when n is fixed. a) n = 10, p = 0.25 b) n = 10, p = 0.5 c) n = 10, p = 0.75 For small samples, binomial distributions are skewed when p is different from 0.5. a) b) c)

13. Hypergeometric distribution ( 超幾何分配 )  A population has two objects (black and white balls, for example), we have N object and M another object (N black balls and M white balls). For a simple random sample without replacement for n times, the number of assigned object (black balls, for example) is a hypergeometric setting.  We can use Bernoulli experiment to understand the hypergeometric setting. A hypergeometric variable X is the sum of dependent bernoulli variables, where Y i =1 if successes and Y i =0 if fails.

14. Hypergeometric distribution  The pdf of a hypergeometric distribution is Parameter a ranges: Where M,N,n are al parameters, M is number of successes, N is number of failures, n is the total trials.  We usually note a hypergeometric distribution as X~ HG (M,N,n)

15. Hypergeometric distribution

16. The Poisson setting  A count X of successes has a Poisson distribution ( 波氏分配 ) in the Poisson setting:  The number of successes that occur in any unit of measure is independent of the number of successes that occur in any non- overlapping unit of measure.  The probability that a success will occur in a time slot is the same for all time slots of equal length and is proportional to the length of the time.  The probability that two or more successes will occur in a unit approaches 0 as the size of the unit becomes smaller. Waiting bus: We count number of buses arriving in a hour, then the number of buses will obey a Possion distribution.

17. Poisson distribution  The distribution of the count X of successes in the Poisson setting is the Poisson distribution with mean λ. The parameter λ is the mean number of successes per unit of measure.  The possible values of X are the whole numbers 0, 1, 2, 3, …. If a is any whole number 0 or greater, then P(X = a) = (e -λ λ a )/a!  The standard deviation of the distribution is the square root of λ.  The Possion is a discrete distribution.

18. Poisson distribution  Let 1.6 be the average number of flaws in a square yard of carpet. The count of X flaws per square yard of carpet is modeled by the Poisson distribution with λ = 1.6.  The probability of two or less flaws per square yard is P(X < 2).  P(X < 2) = P(X = 0) + P(X = 1) + P(X = 2) = (e -1.6 (1.6) 0 )/0! + (e -1.6 (1.6) 1 )/1! + (e -1.6 (1.6) 2 )/2! = 0.2019 + 0.3230 +.02584 = 0.7833

19. Poisson distribution  Two independent Possion variables X and Y with parameters λ 1 and λ 2, we know X+Y is a Possion with mean: λ 1 + λ 2. .  For a binomial distribution, when n approach infinite and p approach 0, the binomial approaches a Possion.

20. Exponential distribution ( 指數分配 )  An exponential distribution describes the time span between two events.  Let X is the time span between two A events, given t >0, where {X > t} means next event A will occur later than time t. Waiting bus: We count waiting time of bus, then the waiting time for next bus will obey a exponential distribution.

21. Exponential distribution  For a Possion, the parameter λ means the expected times of outcomes in a period of time t. So, let t/ λ =β is the expected time span of two lead-lag events.  The pdf of an exponential distribution is  We usually note an exponential variable as X~Exp(β) 。  If X~Exp(β), then E(X)= β, var(X)= β 2

22. Normal distribution  The density function of a normal distribution is  We can describe a normal by two parameters only: mean and variance.

23. Normal distribution feature  The probability density function is in continuous form, and also termed as Gaussian distribution ( 高斯分配 ).  A normal has mean=median=mode.  A normal is in bell shaped, which is also symmetric.  A normal distribution with high standard deviation has fat tail in both sides.  We usually use and to stand for pdf and cdf of a normal. . A normal distribution with mean 0 and standard deviation (or variance) 1 is termed standard normal distribution ( 標準常態分配 ).

24. Normal approximation Back to binomial distribution, if n is large, and p is not too close to 0 or 1, the binomial distribution can be approximated by the normal distribution N(µ  = np, σ  = np(1  p)) Practically, the Normal approximation can be used when both np ≥10 and n(1  p) ≥10. If X is the count of successes in the sample, the sampling distributions for large n is: X approximately N(µ = np, σ 2 = np(1 − p))

25. Normal approximation If 605 of people claim to be satisfied with their lives, what is the probability that at least 610 people out of 1010 surveyed will say they are satisfied with their lives? µ  = np = (1010)(0.60) = 606 σ = √(np(1  p)) = √(1010(0.60)(0.40)) = 15.57 So X is N(606, 15.57); Note: np ≥10 and n(1  p) ≥10. P(X > 610) = P((X – 606)/15.57 > (610 – 606)/15.57) = P(Z > 0.26) = 0.3974

26. Chi-square distribution ( 卡方分配 )  A random variable X obeying Chi-square distribution with degree of freedom 1 is the square of a standard normal Z: Z~ N (0,1) ， then Z 2 ~  2 (1).  Random variable X with degree of freedom n, then we usually express the Chi-square X ~  2 (n).  For Z 1,….Z n being n independent standard normal variables, then X=Z 1 2 +…..+Z n 2 ~  2 (n)  If X ~  2 (n) and Y ~  2 (m) are independent, then X+Y ~  2 (n+m) 。

27. Chi-square distribution   2 (n) random variable X, its mean is n and its variance is 2n.  When degree of freedom increases, Chi-square approaches a symmetric distribution.

28. t distribution (t 分配 )  For Z ~ N (0,1) and Y ~  2 (n) where Z and Y are independent, then t distribution with degree of freedom denotes as t (n) 。  The moment of a t (n) depends on its degree of freedom n.  t (1) has infinite expected value, which is also known Cauchy distribution ( 柯西分配 ). In other words, only when n >1, t (n) has finite expected value.  t (2) has no valid variance. Only when n >2, t (n) has finite variance equal to n/(n – 2).

29. t distribution  When n approaches infinite, the variance of t (n) is closed to 1, meaning that t(n) approaches standard normal.  When t distribution has low df (e.g., n =2 或 4), we will get fat-rail ( 厚尾 ) and more outliers.

30. F distribution (F 分配 )  For two independent Chi-square variables Z ~  2 (m) and Y ~  2 (n) where m and n are df of numerator and denominator.  If X ~ F (m,n), then 1/ X ~ F (n,m)  The moments of F distribution depend on df. When n >2, F (m,n) has mean: n/(n – 2); when n > 4, F (m,n)has variance: