Some Special Discrete Distributions

Transcription

1 Mathematics Department De La Salle University Manila February 6, 2017

2 Some Discrete Distributions Often, the observations generated by different statistical experiments have the same general type of behaviour. Consequently, discrete random variables associated with these experiments can be described by essentially the same probability distributions and therefore can be represented by a single formula. The following discrete distributions will be discussed: Discrete Uniform Binomial Hypergeometric Negative Binomial Geometric Poisson

3 Discrete Uniform Distribution Definition If X is a discrete random variable that could assume any of the values x 1, x 2,..., x k with equal probabilities, then X has the discrete uniform distribution usually denoted by X Uni(k) with probability mass function { 1 f (x) = k if x = x 1, x 2,..., x k 0 otherwise The mean and variance of a discrete uniform random variable is given by µ X = E[X ] = 1 k x i, σx 2 k = Var[X ] = 1 k (x i µ X ) 2 k i=1 i=1

4 Discrete Uniform Distribution-Examples When a die is tossed, each element of the sample space S = {1, 2, 3, 4, 5, 6} occurs with probability 1 6. Therefore, we have a uniform distribution, with X Uni(6). Also, E[X ] = 1 6 ( ) = 3.5 and σ2 X Suppose that an employee is selected at random from a staff of 10 to supervise a certain project. Each employee has the same probability of being selected. If we assume that the employees have been numbered in some way from 1 to 10, the distribution is uniform with Y Uni(10). We note that it is not difficult to see that the distribution of all possible subsets of size n from a finite sample space of size N is itself uniform. Assuming that each subset has an equal chance of being selected. From this, we see that k = ( ) N n

5 Binomial Distribution A binomial experiment is one that possesses the following properties: The experiment consists of n repeated trials. Each trial results in an outcome that may be classified as a success or a failure. The probability of success, denoted by p, remains constant from trial to trial. The repeated trials are independent. Definition The number X of successes in n trials in a binomial experiment is called a binomial random variable with probability distribution denoted by Bin(n, p) with probability mass function given by f (x) = ( n x) p x (1 p) n x, x = 0, 1, 2,..., n. The binomial random variable X has mean µ X = np and variance σx 2 = np(1 p).

6 Binomial Distribution Note If X Bin(1, p), then it is called a Bernoulli random variable. Example Harris, a basketball fanatic, has a shooting average of 60%. He attempts to shoot the ball 12 times. a) What is the probability that he will make exactly eight shots? b) What is the probability that he will make more than one shot? c) What is the probability that he will make between three to seven shots? d) On the average, how many times will Harris make a shot?

7 Binomial Distribution Example 1 A multiple choice quiz has 15 questions, each with 4 possible answers of which only 1 is correct. What is the probability that sheer guesswork yields from 5 to 10 correct answers? 2 The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive? 3 A study conducted at GW University and the National Institute of Health examined national attitudes about tranquilizers. The study revealed that approximately 70% believe that tranquilizers do not really cure anything, they just cover up the real trouble. According to this study, what is the probability that at least 3 of the next 5 people selected at random will be of the opinion that tranquilizers actually do cure the problem rather than just cover it up?

8 Hypergeometric Distribution A hypergeometric experiment is one that possesses the following properties: 1. A random sample of size n is selected from a population of size N. 2. K of the N elements in the population are classified as successes and N K are classified as failures. Definition The number X of successes in a hypergeometric experiment is called a hypergeometric random variable. The probability ( mass K )( N K ) x n x function of this random variable is given by f (x) = ( N n) with mean and variance µ X = nk N, σ2 X = nk N ( 1 K N ) ( N n N 1 )

9 Hypergeometric Distribution-Examples Example 1 If seven cards are dealt from an ordinary deck of 52 playing cards, what is the probability that a. exactly 2 of them will be face cards? b. at least 1 of them will be a queen? 2 Five missiles are selected at random and fired from a depot containing 12 missiles. If there are 3 defective missiles that will not fire, what is the probability that a. all 5 will fire? b. at most 2 will not fire?

10 Negative Binomial Distribution Definition If the random variable X is defined as the number of independent and identical Bernoulli trials at which the k th success occurs, then X has the negative binomial distribution denoted by X NegBin(k, p), with probability mass function given by ( ) x 1 f (x) = p k (1 p) x k k 1 where x = k, k + 1,.... The mean and variance of this random variable is given by µ X = k p, σ2 X = k(1 p) p 2

11 Negative Binomial Example Example Find the probability that a person tossing 3 coins will get either all head or all tails for the second time on the fifth toss. SOLUTION: Using the negative binomial distribution, we see that x = 5, k = 2 and p = 1 4. We want to compute for P(X = x) = P(X = 5) = NegBin(5, 1 4 ) =

12 Geometric Distribution Definition If the random variable X is defined as the number of independent and identical Bernoulli trials at which the first success occurs, then X has the geometric distribution, denoted by X Geo(p) NegBin(k = 1, p). The probability mass function of this random variable is given by f (x) = p(1 p) x 1, x = 1, 2, 3,... with mean and variance µ X = 1 p, σ2 X = 1 p p 2.

13 Geometric Distribution-Example 1 Find the probability that a person flipping a balanced/fair coin 1 requires 4 tosses to get a head. ANS: 16 2 Oyelle plays a shooting game of basketball in the carnival fair during a town fiesta. He hopes to win the prize, a teddy bear, for his girlfriend. He is required to shoot the ball in any one of three attempts. Oyelles shooting average is a poor 20%. What is the probability that he will win the prize on his third attempt? ANS: 0.128

14 Poisson Distribution A process that examines the number of times an event will occur over a specified time interval or region of space is called a Poisson experiment. The number of occurrences of such an event within a specified time interval or region of space is independent of its occurrences in another time interval or region of space. Furthermore, it is nearly impossible for at least two such events to occur simultaneously. A Poisson experiment has the following characteristics: The occurrences (called Poisson events) within a specified time interval or region are independent of the occurrences in the other time interval or region. The number of occurrences is proportional to the length of the time interval or size of the region. The probability of two or more occurrences within a very short time interval or small region is negligible.

15 Poisson Distribution A random variable X defined as the number of occurrences of a Poisson event within a specified time interval or size of a region has the Poisson distribution, denoted by X Poi(λ), and the following probability mass function: f (x) = 3 λ λ x, where x = 0, 1, 2... x! and λ denotes mean rate of occurrence of the Poisson event within the specified time interval or size of the region.

16 Poisson Distribution-Example Mario hired an encoder for the manuscript of the school play he is writing. The encoder makes an average of three errors per page. Assume that the number of errors per page has the Poisson distribution. If a page is randomly selected from the manuscript, find the probability that 1. no error will be found. 2. at least one error will be found. 3. exactly one error will be found

17 Poisson Distribution-Example 1 The average number of days school is closed due to flood during the rainy season in a city in the Region VI is 4. What is the probability that the schools in this city will close for 6 days during a rainy season? ANS: The average number of field mice per acre in a 5-acre wheat field is estimated to be 10. Find the probability that a given acre contains more than 15 field mice? ANS: A restaurant prepares a tossed salad containing on the average 5 vegetables. Find that probability that the salad contains more than 5 vegetables on a given day.

18 Some Exercises 1. By investing in a particular stock, a person can make a profit in 1 year of $4000 with probability 0.4 or take a loss of $1000 with a probability of 0.7. What is the persons expected gain? 2. Find the expected number of jazz records when 4 records are selected at random from a collection of 5 jazz records, 2 classical records and 3 pop records. 3. In a gambling game a man is paid $3 if he draws a jack or a queen and $5 if he draws a king or an ace from an ordinary deck of 52 cards. If he draws any other card, he loses. How much should he pay to play the game if the game is fair?

19 Some Exercises 4. Suppose that on the average 1 in every 1000 is an alcoholic. Find the probability that a random sample of 8000 people will yield fewer than 7 alcoholic? 5. A certain area of the eastern U.S. is, on the average, hit by 6 hurricanes a year. Find the probability that in a given year this area will be hit by fewer than 4 hurricanes. 6. A nationwide survey of 17,000 seniors in a certain university reveals that almost 70% disapprove of daily consumption of a glass of wine according to a report of the university paper. If 18 of these seniors are selected at random and asked their opinions, what is the probability that more than 8 but less than 14 disapprove of a daily consumption of a glass of wine?