Statistics Lecture Notes Dr. Halil İbrahim CEBECİ Chapter 06 Random Variables and Discrete Probability Distributions

Random Variable  Represents a possible numerical value from a random event  Takes on different values based on chance Random Variable Statistics Lecture Notes – Chapter 06 Random Variables Discrete Random Variable Continuous Random Variable

Discrete Random Variable A discrete random variable is a variable that can assume only a countable number of values Many possible outcomes:  number of complaints per day  number of TV’s in a household  number of rings before the phone is answered Only two possible outcomes:  gender: male or female  defective: yes or no  spreads peanut butter first vs. spreads jelly first Discrete Random Variable Statistics Lecture Notes – Chapter 06

Continuous Random Variable A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values)  thickness of an item  time required to complete a task  temperature of a solution  height, in inches These can potentially take on any value, depending only on the ability to measure accurately. Continuous Random Variable Statistics Lecture Notes – Chapter 06

Discrete Probability Distributions Statistics Lecture Notes – Chapter 06 Experiment: Toss 2 Coins. Let x = # heads. T T 4 possible outcomes T T H H HH Probability Distribution x x Value Probability 0 1/4 = /4 = /4 = Probability

Discrete Probability Distributions Statistics Lecture Notes – Chapter 06

Expected Value Statistics Lecture Notes – Chapter 06

Laws of Expected Value Statistics Lecture Notes – Chapter 06

Varriance and Standart Deviation Statistics Lecture Notes – Chapter 06

Laws of Varriance Statistics Lecture Notes – Chapter 06

Probability Distributions Statistics Lecture Notes – Chapter 06 Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal Uniform Exponential

Binomial Distribuiton Statistics Lecture Notes – Chapter 06

Binomial Distribuiton Statistics Lecture Notes – Chapter 06  A manufacturing plant labels items as either defective or acceptable  A firm bidding for a contract will either get the contract or not  A marketing research firm receives survey responses of “yes I will buy” or “no I will not”  New job applicants either accept the offer or reject it

Binomial Distribuiton Statistics Lecture Notes – Chapter 06 The binomial random variable counts the number of successes in n trials of the binomial experiment. It can take on values from 0, 1, 2, …, n. Thus, its a discrete random variable. P(x) n x ! nx pq x n x ! ()!   

Binomial Distribuiton Statistics Lecture Notes – Chapter 06

Binomial Distribuiton Statistics Lecture Notes – Chapter 06 c.What is the probability that Past pass the exam? (minimum %50 requried) P(X ≤ 4) =.967

Binomial Distribuiton Statistics Lecture Notes – Chapter 06 d.What is the possibility that pat gets two answers correct? Find this probability with using Binom tables P(X = 2) = P(X≤2) – P(X≤1) =.678 –.376 =.302

Binomial Distribuiton Statistics Lecture Notes – Chapter 06

Poisson Distribuiton Statistics Lecture Notes – Chapter 06  Named for Simeon Poisson, the Poisson distribution is a discrete probability distribution and refers to the number of events (a.k.a. successes) within a specific time period or region of space  The number of cars arriving at a service station in 1 hour. (The interval of time is 1 hour.)  The number of flaws in a bolt of cloth. (The specific region is a bolt of cloth.)  The number of accidents in 1 day on a particular stretch of highway. (The interval is defined by both time, 1 day, and space, the particular stretch of highway.)

Poisson Distribuiton Statistics Lecture Notes – Chapter 06 Like a binomial experiment, a Poisson experiment has four defining characteristic properties: 1.The number of successes that occur in any interval is independent of the number of successes that occur in any other interval. 2.The probability of a success in an interval is the same for all equal-size intervals 3.The probability of a success is proportional to the size of the interval. 4.The probability of more than one success in an interval approaches 0 as the interval becomes smaller.

Poisson Distribuiton Statistics Lecture Notes – Chapter The Poisson random variable is the number of successes that occur in a period of time or an interval of space in a Poisson experiment. E.g. On average, 96 trucks arrive at a border crossing every hour. E.g. The number of typographic errors in a new textbook edition averages 1.5 per 100 pages. successes time period successes (?!) interval

Poisson Distribuiton Statistics Lecture Notes – Chapter 06

Poisson Distribuiton Statistics Lecture Notes – Chapter 06 Ex6.3 - The number of typographical errors in new editions of textbooks varies considerably from book to book. After some analysis he concludes that the number of errors is Poisson distributed with a mean of 1.5 per 100 pages. The instructor randomly selects 100 pages of a new book. What is the probability that there are no typos? There is about a 22% chance of finding zero errors

Poisson Distribuiton Statistics Lecture Notes – Chapter 06 there is a very small chance there are no typos

Hypergometric Distribuiton Statistics Lecture Notes – Chapter 06

Hypergometric Distribuiton Statistics Lecture Notes – Chapter 06  Two possible outcomes per trial: success or failure

Hypergometric Distribuiton Statistics Lecture Notes – Chapter 06 Note that,

Exercises Statistics Lecture Notes – Chapter 06 Q6.1 - The owner of a boat used for deep-sea fishing charters advertises that his clients catch at least one large salmon (over 15 pounds) on 60% of the charters. Suppose you book four charters for the coming year. a.Find the probability distribution of X, the number of charters on which you catch at least one large salmon. b.Find the probability of your chance to catch «at least two» large salmon on your 4 trip.

Exercises Statistics Lecture Notes – Chapter 06 Q6.2 - The Globe and Mail (3 September 1987) has reported that 80% of Wall Street firms that speculate on corporate takeovers before they become public also help to underwrite such deals. A random sample of 10 firms that speculate on corporate takeovers is selected a.Find the probability that at least half the firms selected underwrite takeovers. b.Find the probability that at most three of the firms selected do not underwrite takeovers. c.Find the expected value and variance of the number of firms selected that underwrite takeovers.

Exercises Statistics Lecture Notes – Chapter 06 Q6.3 - According to Forbes Magazine, 27% of Americans would like to fly in the Space Shuttle. Three Americans are selected at random and asked if they would like to fly in the Space Shuttle. Find the probability distribution of X, the number of persons selected who respond positively.

Exercises Statistics Lecture Notes – Chapter 06 Q6.4 - Records show that there is an average of three accidents each day in a certain city between 2 and 3 P.M. a.Use the Poisson probability formula to find the probability that there will be exactly one accident between 2 and 3 P.M. on a particular day. b.Use the table of Poisson probabilities to check your answer to part a). c.Find the probability that there will be at least three accidents between 2 and 3 P.M. on a particular day. d.Find the probability that there will be at least three accidents between 2 and 2:30 P.M. on a particular day.

Exercises Statistics Lecture Notes – Chapter 06 Q6.5 - A shipment of 10 items has two defective and eight nondefective items. In the inspection of the shipment, a sample of items will be selected and tested. If a defective item is found, the shipment of 10 items will be rejected. a. If a sample of three items is selected, what is the probability that the shipment will be rejected? b. If a sample of four items is selected, what is the probability that the shipment will be rejected? c. If a sample of five items is selected, what is the probability that the shipment will be rejected? d. If management would like a.90 probability of rejecting a shipment with two defective and eight nondefective items, how large a sample would you recommend?

Exercises Statistics Lecture Notes – Chapter 06 Q6.5 - Blackjack, or twenty-one as it is frequently called, is a popular gambling game played in Las Vegas casinos. A player is dealt two cards. Face cards (jacks, queens, and kings) and tens have a point value of 10. Aces have a point value of 1 or 11. A 52-card deck contains 16 cards with a point value of 10 (jacks, queens, kings, and tens) and four aces. a.What is the probability that both cards dealt are aces or 10-point cards? b.What is the probability that both of the cards are aces? c.What is the probability that both of the cards have a point value of 10?