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Chapter 4 Probability Distributions

Chapter 4 Probability Distributions. 4-1 Random Variables 4-2 Binomial Probability Distributions 4-3 Mean, Variance, Standard Deviation for the Binomial Distribution 4-4 Other Discrete Probability Distributions. Overview. This chapter will deal with the construction of

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Chapter 4 Probability Distributions

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  1. Chapter 4Probability Distributions 4-1 Random Variables 4-2 Binomial Probability Distributions 4-3 Mean, Variance, Standard Deviation for the Binomial Distribution 4-4 Other Discrete Probability Distributions

  2. Overview This chapter will deal with the construction of probability distributions by combining the methods of Chapter 2 with the those of Chapter 3. Probability Distributions will describe what will probably happen instead of what actually did happen.

  3. Combining Descriptive Statistics Methods and Probabilities to Form a Theoretical Model of Behavior

  4. 4-1 Random Variables

  5. Definitions • Random Variable a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure • Probability Distribution a graph, table, or formula that gives the probability for each value of the random variable

  6. Probability DistributionNumber of Girls Among Fourteen Newborn Babies x P(x) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0.000 0.001 0.006 0.022 0.061 0.122 0.183 0.209 0.183 0.122 0.061 0.022 0.006 0.001 0.000 • Questions: • Let X = # of girls • P(X=3) = ? • P(X ≤ 2) = ? • P(X ≥ 1) = ?

  7. Probability Histogram

  8. Definitions • Discrete random variable has either a finite number of values or countable number of values, where ‘countable’ refers to the fact that there might be infinitely many values, but they result from a counting process. • Continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale with no gaps or interruptions.

  9. Requirements for Probability Distribution  P(x) = 1 where x assumes all possible values 0 P(x) 1 for every value of x See #3 on hw

  10. Mean, Variance and Standard Deviation of a Probability Distribution Formula (Mean) µ = [x•P(x)] Formula (Variance) 2= [(x - µ)2 • P(x)] Formula (Standard Deviation) =[(x - µ)2 • P(x)]

  11. Roundoff Rule for µ, 2, and Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of xare integers, round µ, 2, and  to one decimal place.

  12. TI-83 Calculator Calculate Meanand Std. Dev from a Probability Distribution • Press Stat • Press “1” Edit • Enter values of random variable (x) in L1 • Enter probability P(x) in L2 • Press Stat • Cursor over to CALC • Choose the 1-Var stats option • Enter 1-Var stats L1,L2

  13. Using Excel See Probability Distribution Worksheet Examples: • See Introduction to probability distributions handout • Go to Excel (dice example – class assignment) • Find missing probability, mean, SD and unusual values(test question see #6 on hw)

  14. Unusual and Unlikely Values • Unusual if greater than 2 standard deviations from the mean, that is x + 2 and x – 2s • Unlikely if probability is very small, usually less than .05. This is consistent with the 2 idea associated with the empirical rule. #6f and 7c on HW

  15. Definition Expected Value The average value of outcomes E =  [x • P(x)]

  16. E =  [x • P(x)] Example: #9 on hw Event Win Lose x $5 - $5 P(x) 244/495 251/495 x • P(x) 2.465 - 2.535 E = -$.07

  17. 4-2 Binomial Random Variables

  18. Binomial Random Variables Facts: • Discrete (we can count the outcomes) • Have to do with random variables having 2 outcomes. Examples: heads/tails, boy/girl, yes/no, defective/not defective, etc. • A binomial distribution is the sum of several trials. Example: # of heads when a coin is tossed three times

  19. Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials p = probability of success in one of n trials q = probability of failure in one of n trials (q = 1 - p) P(x) = probability of getting exactly x successes among n trials

  20. Method 1 Binomial Probability Formula n! • P(x) = • px• qn-x (n - x )! x! • P(x) = nCx• px•qn-x for calculators use nCr key, where r = x

  21. Binomial Probability Formula n! P(x) = • px•qn-x (n - x )! x! Probability of x successes among n trials for any one particular order Number of outcomes with exactly x successes among n trials

  22. Example: Toss a coin 3 times. • Let x = number of heads and find • P(2) = • P(at least 2) This is a binomial experiment so you need to know 4 things p, q, n and x. p=.5 q=.5 n=3 a) x = 2 b) x = 0 then 1 then 2 On the test you will have to construct the entire probability distribution for tossing a coin “n” times and observing the number of heads. Should do this before the test.

  23. Example: Find the probability of getting exactly 2 correct responses among 5 different requests from directory assistance. Assume in general, they are correct 80% of the time. This is a binomial experiment where: n = 5 x = 2 p = 0.80 q = 0.20 Using the binomial probability formula to solve: P(2) = 5C2• 0.8 • 0.2 = 0.0512 3 2

  24. Method 2 Binomial Table Two tables are available on website

  25. Example: Using Table for n = 5 and p = 0.80, find the following: a) The probability of exactly 2 successesb) The probability of at most 2 successesc) The probability of at least 1 success a) P(2) = 0.0512 b) P(at most 2) = P(0) or P(1) or P(2) = 0.0003 + 0.0064 + 0.0512 = 0.0579 c) P(at least 1) = 1 – P(0) = 1 – .0003 = .9997 Test Question

  26. Method 3 Using Technology • Calculator function (TI-83) • See binomial distribution worksheet • See also coin example worksheet

  27. TI-83 Calculator Finding Binomial Probabilities (complete distribution) • Press 2nd Distr • Choose binopdf • Enter binopdf(n,p) • Press STO L2 (stores probabilities in column L2) • Press Stat • Choose Edit (to view probabilities) • Optional: enter the values of the random variable in L1

  28. TI-83 Calculator Finding Binomial Probabilities (individual value) • Press 2nd Distr • Choose binopdf • Enter binopdf(n,p,x)

  29. TI-83 Calculator Finding Binomial Probabilities (cummulative) • Press 2nd Distr • Choose binocdf • Enter binocdf(n,p,x) This yields the sum of the probabilities from 0 to x. Example: Let n=6 and p=0.2 P(X<3) = binocdf(6,.2,3)

  30. 4.3 Mean, variance and standard deviation of a Binomial Probability Distribution

  31. For Any Discrete Probability Distribution the general formulas are: • µ = [x • P(x)] • 2=  [(x - µ) 2• P(x) ] =  [(x - µ) 2 • P(x)]

  32. For Binomial Distributions: • µ = n • p • 2 = n • p • q = n • p • q

  33. Example: Find the mean and standard deviation for students that guess answers on a multiple choice test with 5 answers and 20 questions. • We previously discovered that this scenario could be considered a binomial experiment where: • n = 20 • p = 0.2 • q = 0.8 • Using the binomial distribution formulas: µ = (20)(0.2) = 4 correct answers = (20)(0.2)(0.8) = 1.8 answers (rounded) Test question

  34. Reminder • Maximum usual values = µ + 2  • Minimum usual values = µ - 2 

  35. Example: Determine whether guessing 7 correct answers is unusual. For this binomial distribution, • µ = 4 answers • = 1.8 answers • µ + 2  = 4 + 2(1.8) = 7.6 • µ - 2  = 4 - 2(1.8) = .4 The usual number of correct answers would be from .4 to 7.6, so guessing 7 correct answers would not be an unusual result. Test question

  36. 4.4 Other Discrete Probability Distributions • Poisson • Geometric • Hypergeometric • Negative Binomial • And more

  37. Poisson Distribution Definition a discrete probability distribution that applies to occurrences of some event over a specific interval. Will be a question on the test for you to differentiate between a binomial and a poisson distribution

  38. µ x• e-µ P(x) = wheree  2.71828 x! Definition Poisson Distribution a discrete probability distribution that applies to occurrences of some event over a specific interval. Probability Formula

  39. Example: Look at #1 • Why is this a poisson distribution? • µ = 5 • We need to find various probabilities using • Let’s find P(7) • Look at the Poisson function in Excel µ x• e-µ P(x) = x!

  40. Geometric Distribution Definition a discrete probability distribution of the number of trials needed to get one success. Will be a question on the test for you to differentiate between a binomial, poisson and a geometric distribution

  41. Geometric Distribution Example: Roll a die 5 times. What is the probability of getting your first 2 on the 5th roll.

  42. Negative Binomial Distribution Definition a discrete probability distribution of the number of trials needed to get a get a specified number of successes.

  43. Negative Binomial Distribution Example a basketball player has a 70% chance of making a free throw, what is the probability of making his 3rd free throw on his 5th shot.

  44. Hypergeometric Distribution Hypergeometric Experiment • A sample of size n is randomly selected without replacement from a population of N items. . In the population, k items can be classified as successes, and N - k items can be classified as failures.

  45. Hypergeometric Distribution Notation • N: The number of items in the population. • k: The number of items in the population that are classified as successes. • n: The number of items in the sample. • x: The number of items in the sample that are classified as successes. • kCx: The number of combinations of k things, taken x at a time.

  46. Hypergeometric Distribution Example Suppose we randomly select 5 cards without replacement from an ordinary deck of playing cards. What is the probability of getting exactly 2 red cards (i.e., hearts or diamonds)? P = [ kCx ] [ N-kCn-x ] / [ NCn ] = [ 26C2 ] [ 26C3 ] / [ 52C5 ] = [ 325 ] [ 2600 ] / [ 2,598,960 ] = 0.32513

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