Lesson 1 Chapter 5 probability

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7th edition Statistics for Management
Inferential Statistics
International Business School, Hanze university of Applied Science, Groningen, The Netherlands

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  • A continuous random variable is one whose values are uncountable. An excellent example of a continuous random variable is the amount of time to complete a task. For example, let X = time to write a statistics exam in a university where the time limit is 3 hours and students cannot leave before 30 minutes. The smallest value of X is 30 minutes. If we attempt to count the number of values that X can take on, we need to identify the next value. Is it 30.1 minutes? 30.01 minutes? 30.001 minutes? None of these is the second possible value of X because there exist numbers larger than 30 and smaller than 30.001. It becomes clear that we cannot identify the second, or third, or any other values of X (except for the largest value 180 minutes). Thus, we cannot count the number of values and X is continuous.
  • http://www.shodor.org/interactivate/activities/ExpProbability/
  • Solution. If we let X denote the number of subscribers who qualify for favorable rates, then X is a binomial random variable with n = 10 and p = 0.70. And, if we let Y denote the number of subscribers who don't qualify for favorable rates, then Y , which equals 10 −  X , is a binomial random variable with n = 10 and q = 1 − p = 0.30. We are interested in finding P ( X ≥ 4). We can't use the cumulative binomial tables, because they only go up to p = 0.50. The good news is that we can rewrite  P ( X  ≥ 4) as a probability statement in terms of Y : P ( X ≥ 4) = P (− X ≤ −4) = P (10 − X ≤ 10 − 4) = P ( Y ≤ 6) Now it's just a matter of looking up the probability in the right place on our cumulative binomial table. To find  P ( Y  ≤ 6), we:  Find  n  = 10  in the first column on the left. Find the column containing  p  = 0.30 . Find the  6  in the second column on the left, since we want to find  F (6) =  P ( Y  ≤ 6). Now, all we need to do is read the probability value where the  p  = 0.30 column and the ( n  = 10, y  = 6) row intersect. What do you get? Do you need a hint? The cumulative binomial probability table tells us that  P ( Y  ≤ 6) =  P ( X  ≥ 4) = 0.9894. That is, the probability that at least four people in a random sample of ten would qualify for favorable rates is 0.9894.
  • Lesson 1 Chapter 5 probability

    1. 1. Statistics 2 Dr. Ning DING [email_address] I.007, IBS Lesson 01, Sept 2011
    2. 2. Table of Contents <ul><li>What is Probability Distribution? </li></ul><ul><li>Random Variables </li></ul><ul><li>Use of Expected Value in Decision Making </li></ul><ul><li>The Binomial Distribution </li></ul><ul><li>The Poisson Distribution </li></ul><ul><li>The Normal Distribution: A Distribution of a Continuous Random Variable </li></ul>Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution
    3. 3. What is a probability distribution? <ul><ul><li>= theoretical frequency distribution </li></ul></ul>AnimationDemo <ul><ul><li>= listing of the probabilities of all the possible outcomes </li></ul></ul>Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution
    4. 4. Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution What is a probability distribution?
    5. 5. Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution What is a probability distribution?
    6. 6. <ul><li>Discrete </li></ul><ul><li>Continuous </li></ul>= only a limited number of values = any value within a given range, e.g. height or weight Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Types of probability distribution?
    7. 7. <ul><li>Can be either discrete or continuous </li></ul>Chapter 5, SC 5-2 P.229 Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Random Variables
    8. 8. <ul><li>What did you expect? </li></ul>Chapter 5, SC 5-2 P.229 Average number of women screened daily is: 108.02 Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution The Expected Value of Random Variables
    9. 9. <ul><li>Combine probabilities and monetary values </li></ul>Chapter 5, SC 5-2 P.229 Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Use of Expected Value in Decision Making
    10. 10. <ul><li>= probability distribution of a discrete random variable </li></ul>Probability of r successes in n trials = p r q n-r factorial ! 阶乘 Fakultät factorial la factorielle Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution The Binomial Distribution n ! r ! ( n-r ) ! Number of trials Number of successes desired Probability of success Probability of failure
    11. 11. <ul><li>Use of the Bernoulli Process </li></ul>The Binomial Distribution Probability of r successes in n trials = p r q n-r 2 tails in 3 tosses? Probability of 2 successes in 3 trials = 0.5 2 *0.5 3-2 0.5 2 *0.5 =0.3750 Example: Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution n ! r ! ( n-r ) ! 3! 2 ! ( 3-2 ) ! 3*2*1 2 *1 * 1*1 !
    12. 12. The Binomial Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution
    13. 13. <ul><li>= for discrete random variables </li></ul>The Poisson Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Mean of successses Probability of x occurences 2.71828 x factorial
    14. 14. The Poisson Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Example: Mean=5 accidents per month; Calculate the probability of exactly 0,1,2,3,or 4 accidents e = 0.00674 P=0.12469 P=0.26511 Step 1: look up in Table 4a to get e Step 2: Calculate the P
    15. 15. Using Table to solve the problem Table 4 (b) The Poisson Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution
    16. 16. The Normal Distribution Characteristics of the Normal Probability Distribution Unimodal Mean at the center Mean=median=mode Tails off indefinitely Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution
    17. 17. The Normal Distribution The same mean Different SD Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution
    18. 18. The Normal Distribution The same SD Different mean Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution
    19. 19. The Normal Distribution The largest SD? The largest mean? Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution
    20. 20. The Normal Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution
    21. 21. The Normal Distribution z score Mean length of time is 500 hours; SD is 100 hours; Q1. What is the probability of >500 hours? Q2. What is the probability of 500~650 hours? Example: Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution 0.50 0.4332
    22. 22. The Normal Distribution z score Mean length of time is 500 hours; SD is 100 hours; Q3. What is the probability of >700 hours? Q4. What is the probability of 550~650 hours? Example: Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution 0.5-0.4772=0.0228 0.4332-0.1915 = 0.2417
    23. 23. The Normal Distribution z score Mean length of time is 500 hours; SD is 100 hours; Q5. What is the probability of <580 hours? Q4. What is the probability of 420~570 hours? Example: Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution 0.2881+0.5=0.7881 0.2580+0.2881=0.5461
    24. 24. The Normal Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution z score Mean of the distribution SD of the distribution value
    25. 25. Summary <ul><li>What is Probability Distribution? </li></ul><ul><li>Random Variables </li></ul><ul><li>Use of Expected Value in Decision Making </li></ul><ul><li>The Binomial Distribution </li></ul><ul><li>The Poisson Distribution </li></ul><ul><li>The Normal Distribution: A Distribution of a Continuous Random Variable </li></ul>
    26. 26. The Normal Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Many utility companies promote energy conservation by offering discount rates to consumers who keep their energy usage below certain established subsidy standards. A recent EPA report notes that 70% of the island residents of Puerto Rico have reduced their electricity usage sufficiently to qualify for discounted rates. If ten residential subscribers are randomly selected from San Juan, Puerto Rico, what is the probability that at least four qualify for the favorable rates? Test yourself Extra Reading
    27. 27. The Binomial Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution <ul><li>Example: Pat Statsdud is a student taking a statistics course. Unfortunately, Pat is not a good student. Pat does not read the textbook before class, does not do homework, and regularly misses class. Pat intends to rely on luck to pass the next quiz. The quiz consists of 10 multiple-choice questions. Each question has five possible answers, only one of which is correct. Pat plans to guess the answer to each question. </li></ul><ul><ul><li>a. What is the probability that Pat gets no answers correct? </li></ul></ul>P (0) = .1074 Example: Extra Reading Probability of r successes in n trials = n ! r ! ( n-r ) ! p r q n-r Probability of 0 successes in 10 trials = 10 ! 0 ! ( 10-0 ) ! 0.2 0 0.8 10-0
    28. 28. <ul><li>Example: Pat Statsdud is a student taking a statistics course. Unfortunately, Pat is not a good student. Pat does not read the textbook before class, does not do homework, and regularly misses class. Pat intends to rely on luck to pass the next quiz. The quiz consists of 10 multiple-choice questions. Each question has five possible answers, only one of which is correct. Pat plans to guess the answer to each question. </li></ul><ul><ul><li>b. What is the probability that Pat gets two answers correct? </li></ul></ul>P (2) = .3020 The Binomial Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Example: Extra Reading Probability of r successes in n trials = n ! r ! ( n-r ) ! p r q n-r Probability of 2 successes in 10 trials = 10 ! 2 ! ( 10-2 ) ! 0.2 2 0.8 10-2
    29. 29. Chapter 5, SC 5-2 P.229 $20 per case $30 per case Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Combining Probabilities and Monetary Values Example: Extra Reading
    30. 30. <ul><ul><li>Obsolescence Losses </li></ul></ul><ul><ul><li>Opportunity Losses </li></ul></ul>Chapter 5, SC 5-2 P.229 Stock too much Stock too few Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Combining Probabilities and Monetary Values Extra Reading
    31. 31. Chapter 5, SC 5-2 P.229 Obsolescence: Stock too much Opportunity: Stock too few $20 per case $30 per case 30 30 30 60 90 60 20 20 20 40 40 60 Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Combining Probabilities and Monetary Values Extra Reading Possible Requests for Strawberries Possible Stock Options 10 11 12 13 10 0 20 40 60 11 30 0 20 40 12 60 30 0 20 13 90 60 30 0
    32. 32. Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Combining Probabilities and Monetary Values Chapter 5, SC 5-2 P.229 10 11 12 13
    33. 33. The Poisson ~Binomial Distribution When n >=20 and p =< 0.05. Equation: P.254 Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Extra Reading
    34. 34. µ = n * p Mean of a binomial distribution σ = sqrt(npq) Standard deviation of a binomial distribution Central Tendency and Dispersion of Binomial Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution Extra Reading
    35. 35. Central Tendency and Dispersion of Binomial Distribution Probability Distribution? Random Variables Expected Value in Decision Making The Binomial Distribution The Poisson Distribution The Normal Distribution 20% defective packages 10 packages What are the mean and standard deviation? Example: µ = n * p =10*0.2 =2 σ = sqrt(npq) =sqrt (10*0.2*0.8) =1.265 Extra Reading

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