1. 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Discrete Distributions Chapter 5

2. 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Random Variables  Random variable a variable (typically represented by x) that takes a numerical value by chance.  For each outcome of a procedure, x takes a certain value, but for different outcomes that value may be different. pg. 205

3. 3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Examples:  Number of boys in a randomly selected family with three children. Possible values: x=0,1,2,3  The weight of a randomly selected person from a population. Possible values: positive numbers, x>0

4. 4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Examples:  Genetics: If two pea plants have both green and yellow pod genes, the probability that offspring pea plants has a green pod is.75 That is P(green) =.75

5. 5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Discrete and Continuous Random Variables  Discrete random variable Either a finite number of values or countable number of values (resulting from a counting process)  Continuous random variable Infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions pg. 206

6. 6 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Probability Distributions  Probability distribution A description that gives the probability for each value of the random variable; often expressed in the format of a table, graph, or formula Pea pod histogram on page 207

7. 7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. xP(x) 01/8 13/8 2 31/8 Tables Values: Probabilities:

8. 8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Graphs The probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities.

9. 9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Requirements for Probability Distribution ∑ P ( x ) = 1 where x assumes all possible values. 0  P ( x )  1 for every individual value of x. pg. 207

10. 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mean, Variance and Standard Deviation of a Probability Distribution µ = ∑ [ x P(x) ] Mean σ 2 = ∑ [ (x – µ) 2 P(x )] Variance σ 2 = ∑ [ x 2 P(x )] – µ 2 Variance (shortcut) σ = ∑ [ x 2 P(x )] – µ 2 Standard Deviation

11. 11 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Press the STAT button and choose EDIT Enter the x-values into the list L 1 and the P(x) values into the list L 2 Press the STAT button and choose CALC Choose 1-Var Stats and press ENTER Type in L 1 then, (comma) then L 2 on that line, you will see 1-Var Stats L 1,L 2 Press ENTER You will see x-bar=…, it is actually m (mean) and sx=…, it is actually s (st. deviation) Using TI-83/84 calculator

12. 12 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Finding the Mean, Variance, and Standard Deviation Example 5, page 209 Table 5-1 describes the probability distribution for the number of peas with green pods among five offspring peas obtained from parents both having the green/yellow pair of genes.

13. 13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using Excel - Mean Put the x-values in column 1 Put the P(x)-values in column 2 In column 3 enter “=A1*B1” Copy and paste that cell to the entire column of data At the bottom of column 3 enter “=sum(c1:cN)” - N is the last row of the data Table 5-3, pg 209

14. 14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using Excel - Mean

15. 15 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using Excel – Standard Deviation Continuing In column 4 enter “=power(A1-$c$M,2)*B1” Copy and paste that cell to the entire column of data At the bottom of column 4 enter “=sum(d1:dN)” - N is the last row of the data

16. 16 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using Excel – Standard Deviation

17. 17 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. More detailed example Using Excel to find the mean and standard deviation of a discrete probability distribution

18. 18 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (1)Enter the values and their probabilities as separate columns Using Excel to find the mean and standard deviation of a discrete probability distribution

19. 19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (2)Check to make sure the values of P(x) add up to 1: Using Excel (continued)

20. 20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (3)In column C, for each value of x, calculate x*P(x): Using Excel (continued)

21. 21 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (4)The sum of column C will give the value of the mean, μ Using Excel (continued)

22. 22 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (5)In column D, calculate the values of X 2 *P(x) Using Excel (continued)

23. 23 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (6)Calculate the sum of the squares times probability - X 2 *P(x) Using Excel (continued)

24. 24 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (7)Use the shortcut formula for the variance: Using Excel (continued)

25. 25 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (8)Finally, the Standard deviation is just the square root of the variance: Using Excel (continued)

26. 26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. How to Choose Lottery Numbers Note the sidebar on page 209 about choosing numbers in a lottery.

27. 27 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ, σ, and σ 2 to one decimal place. Pg 210 Roundoff Rule for µ, σ, and σ 2

28. 28 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Identifying Unusual Results Range Rule of Thumb According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits: Maximum usual value = μ + 2σ Minimum usual value = μ – 2σ

29. 29 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Identifying Unusual Results By Probabilities Using Probabilities to Determine When Results Are Unusual:  Unusually high: a particular value x is unusually high if P(x or more) ≤ 0.05.  Unusually low: a particular value x is unusually low if P(x or fewer) ≤ 0.05.

30. 30 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Uniform Probability Distribution A uniform probability distribution has the following property: P(x) = c The value c is a constant, so every event is just a likely as every other event. If there are n events P(x) = 1n1n

31. 31 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Binomial Probability Distribution A binomial probability distribution results from a procedure that meets all the following requirements: 1. The procedure has a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure). 4. The probability of a success remains the same in all trials. pp 218-219

32. 32 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Notation for Binomial Probability Distributions S and F (success and failure) denote the two possible categories of all outcomes; p and q denote the probabilities of S and F, respectively: P(S) = p(p = probability of success) P(F) = 1 – p = q(q = probability of failure) Note that “success” may not be desirable.

33. 33 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Notation (continued) n denotes the fixed number of trials. x denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive. p denotes the probability of success in one of the n trials. q denotes the probability of failure in one of the n trials. P(x) denotes the probability of getting exactly x successes among the n trials. pg 219

34. 34 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Methods for Finding Probabilities We will now discuss three methods for finding the probabilities corresponding to the random variable x in a binomial distribution.

35. 35 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Method 1: Using the Binomial Probability Formula P(x) = p x q n-x ( n – x )! x ! n !n ! for x = 0, 1, 2,..., n where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p) pg 221

36. 36 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Rationale for the Binomial Probability Formula P(x) = p x q n-x n !n ! ( n – x )! x ! The number of outcomes with exactly x successes among n trials

37. 37 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Binomial Probability Formula P(x) = p x q n-x n !n ! ( n – x )! x ! Number of outcomes with exactly x successes among n trials The probability of x successes among n trials for any one particular order

38. 38 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Method 2: Using TI-83/84 Press 2 nd VARS to get the DISTR menu Scroll down to binomialpdf( and press ENTER Type in three values: n, p, x (separated by commas) and close the parenthesis You see a line like binomialpdf(10,.3,6) Press ENTER and read the probability of the value x (successes) in n trials

39. 39 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Alternative use of TI-83/84 Press 2 nd VARS to get the DISTR menu Scroll down to binomialcdf( and press ENTER Type in three values: n, p, x (separated by commas) and close the parenthesis You see a line like binomialcdf(10,.3,6) Press ENTER and read the combined probability of all values from 0 to x (i.e., probability that there are at most x successes)

40. 40 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Method 3: Excel Use an Excel canned function

41. 41 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (1)With the cell you want highlighted, From the home tab, click the arrow beside the ∑ button, and select “more functions” Using Excel (2007 or later) for Binomial Distributions

42. 42 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using Excel (2007 or later) for Binomial Distributions (2)In the pop-up window, scroll down and select the category “statistical”

43. 43 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using Excel (2007 or later) for Binomial Distributions (3)Under “select a function” scroll down to BINOMDIST and click “ok”

44. 44 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using Excel (2007 or later) for Binomial Distributions (4)Fill in the values required Number_s # of successes Trials # of trials Probability_s probability of success Cumulative TRUE if you want the probability of x or fewer successes, FALSE if you just want the probability of exactly x successes.

45. 45 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using Excel (2007 or later) for Binomial Distributions (5)Now look back at the cell you had highlighted: From now on, you can just enter this short form if you find it quicker.

46. 46 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. An “unfair coin” has a 0.55 probability of getting heads and is tossed 10 times Example What is the probability of getting exactly 5 heads? What is the probability of getting at least 4 heads?

47. 47 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Probability of heads: p = 0.55 Number of tosses: n = 10 “Exactly 5 heads” → P(5) P(5) = 0.234 Example

48. 48 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Probability of heads: p = 0.55 Number of tosses: n = 10 “at most 4 heads” → ∑P(4) ∑P(4) = 0.262 Example

49. 49 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. An “unfair coin” has a 0.55 probability of getting heads and is tossed 10 times Example What is the probability of getting exactly 5 heads? What is the probability of getting at least 4 heads? p = 0.55n = 10 P(5) = 0.234 ∑P(4) = 0.262

50. 50 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Binomial Distribution: Formulas Std. Dev.  σ  = n p q Mean µ = n p Variance σ 2  = n p q Where n = number of fixed trials p = probability of success in one of the n trials q = probability of failure in one of the n trials

51. 51 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Interpretation of Results Maximum usual values = µ + 2  Minimum usual values = µ – 2  It is especially important to interpret results. The range rule of thumb suggests that values are unusual if they lie outside of these limits: