1. Statistics for Business and Economics
Chapter 4
Random Variables & Probability Distributions

2. Learning Objectives
Distinguish Between the Two Types of Random Variables
Describe Discrete Probability Distributions
Describe the Uniform and Normal Distributions
As a result of this class, you will be able to...

3. Learning Objectives (continued)
Explain Sampling Distributions
Solve Probability Problems Involving Sampling Distributions

4. Types of Random Variables

5. Data Types
Data
Quantitative
Qualitative
Continuous
Discrete

6. Discrete Random Variables

7. Data Types
Data
Quantitative
Qualitative
Continuous
Discrete

8. Discrete Random Variable
A numerical outcome of an experiment
Example: Number of tails in 2 coin tosses
Discrete random variable
Whole number (0, 1, 2, 3, etc.)
Obtained by counting
Usually a finite number of values
Poisson random variable is exception ()

9. Discrete Random Variable Examples
Possible Values
Experiment
Make 100 Sales Calls
# Sales
0, 1, 2, ..., 100
Inspect 70 Radios
# Defective
0, 1, 2, ..., 70
Answer 33 Questions
# Correct
0, 1, 2, ..., 33
Count Cars at Toll
Between 11:00 & 1:00
# Cars
Arriving
0, 1, 2, ..., ∞

10. Continuous Random Variables

11. Data Types
Data
Quantitative
Qualitative
Continuous
Discrete

12. Continuous Random Variable
A numerical outcome of an experiment
Weight of a student (e.g., 115, 156.8, etc.)
Continuous Random Variable
Whole or fractional number
Obtained by measuring
Infinite number of values in interval
Too many to list like a discrete random variable

13. Continuous Random Variable Examples
Possible Values
Experiment
Weigh 100 People
Weight
45.1, 78, ...
Measure Part Life
Hours
900, 875.9, ...
Amount spent on food
$ amount
54.12, 42, ...
Measure Time
Between Arrivals
Inter-Arrival
Time
0, 1.3, 2.78, ...

14. Probability Distributions for Discrete Random Variables

15. Discrete Probability Distribution
List of all possible [x, p(x)] pairs
x = value of random variable (outcome)
p(x) = probability associated with value
Mutually exclusive (no overlap)
Collectively exhaustive (nothing left out)
0  p(x)  1 for all x
 p(x) = 1

16. Discrete Probability Distribution Example
Experiment: Toss 2 coins. Count number of tails.
Probability Distribution
Values, x Probabilities, p(x)
0 1/4 = .25
1 2/4 = .50
2 1/4 = .25
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17. Visualizing Discrete Probability Distributions
Listing
Table
{ (0, .25), (1, .50), (2, .25) }
f(x)
p(x)
# Tails
Count 1
.25 1 2
.50
Experiment is tossing 1 coin twice.
Graph 2 1
.25
p(x)
.50
Formula
.25 x n !
.00 p ( x ) =
px(1 – p)n - x 1 2
x!(n – x)!

18. Summary Measures Expected Value (Mean of probability distribution)
Weighted average of all possible values
 = E(x) = x p(x)
Variance
Weighted average of squared deviation about mean
2 = E[(x (x p(x)
population notation is used since all values are specified.
Standard Deviation ●

19. Summary Measures Calculation Tablex
p(x)
x p(x)
x – 
(x – )2
(x – )2p(x)
Total
xp(x)
(x p(x)

20. Thinking Challenge
You toss 2 coins. You’re interested in the number of tails. What are the expected value, variance, and standard deviation of this random variable, number of tails?
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21. Expected Value & Variance Solution*
p(x)
x p(x)
x – 
(x – ) 2
(x – ) 2p(x)
.25
.50 
= 1.0
-1.00
1.00
.25
2 = .50
 = .71 1
.50 2
.25
1.00
1.00

22. Portfolio Selection Case

23. Portfolio Selection Case

24. Data Types
Data
Quantitative
Qualitative
Continuous
Discrete

25. Probability Distributions for Continuous Random Variables

26. Continuous Probability Density Function
Mathematical formula
Shows all values, x, and frequencies, f(x)
f(x) Is Not Probability
Value
(Value, Frequency)
Frequency
f(x) a b x
(Area Under Curve) f x dx ( )
All x a b     1 0,
Properties

27. Continuous Random Variable Probability( a  x  b )   f ( x ) dx
Probability Is Area Under Curve! a
f(x) x a b
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28. Continuous Probability Distributions
Uniform
Normal

29. Uniform Distribution x f(x) d c a b 1. Equally likely outcomes
2. Probability density function
3. Mean and Standard Deviation

30. Uniform Distribution Example
You’re production manager of a soft drink bottling company. You believe that when a machine is set to dispense 12 oz., it really dispenses 11.5 to 12.5 oz. inclusive. Suppose the amount dispensed has a uniform distribution. What is the probability that less than 11.8 oz. is dispensed?
SODA

31. Uniform Distribution Solution
f(x)
1.0 x
11.5
11.8
12.5
P(11.5  x  11.8) = (Base)(Height)
= ( )(1) = .30

33. Normal Distribution

34. Importance of Normal Distribution
Describes many random processes or continuous phenomena
Can be used to approximate discrete probability distributions
Example: binomial
Basis for classical statistical inference

35. Normal Distribution f ( x ) x ‘Bell-shaped’ & symmetrical
Mean, median, mode are equal
IQR is 1.33
Random variable has infinite range f ( x ) x
Mean Median Mode

36. Probability Density Function
f(x) = Frequency of random variable x
 = Population standard deviation
 = ; e =
x = Value of random variable (– < x < )
 = Population mean

37. Effect of Varying Parameters ( & )
f(X) B A C X

38. Normal Distribution Probability
Probability is area under curve! f ( x )
Use JMP!!! x c d

39. Normal Distribution Thinking Challenge
You work in Quality Control for GE. Light bulb life has a normal distribution with = 2000 hours and = 200 hours. What’s the probability that a bulb will last:
A. Between 2100 and 2400 hours?
Less than 1470 hours?
More than 2500 hours
Greater than 2000 hours
Allow students about minutes to solve this.

40. Using JMP for Normal Probabilities
1. From JMP, open the file “Distribution_Calculator.jsl”
Edit >> Run Script
Now
1. Fill in the mean and SD
2. Click type of calculation
3. Click type of probability
4. Give boundary value(s)
5. Click Show Values

41. Using JMP for Normal Probabilities
Part (a) solution:

42. Finding Normal Percentiles
The .35 percentile:
Find the X-value such that 35% of the population falls below this value and 65% fall above it.
.35
.65 X
This is just the opposite of finding normal probabilities:
Before: Given X value(s), find the probability
Now: Given a tail probability, find the X value
In JMP, just click the “Input probability and calculate values” button

43. Problem: Find the .35 percentile for a normal distribution with mean 20 and standard deviation 5.
Click here

44. Reliability Example
Life testing has revealed that a particular type of TV picture tube has a length of life that is approximately normally distributed with a mean of 8000 hours and a standard deviation of 1000 hours. The manufacturer wants to set a guarantee period for the tube that will obligate the manufacturer to replace no more than 5% of all tubes sold. How long should the guarantee period be?

45. Assessing Normality

46. Assessing Normality
Draw a histogram or stem–and–leaf display and note the shape
Compute the intervals x + s, x + 2s, x + 3s and compare the percentage of data in these intervals to the Empirical Rule (68%, 95%, 99.7%)
Calculate
If ratio is close to 1.3, data is approximately normal

47. Assessing Normality Continued
Draw a Normal Probability Plot
Observed value
Expected Z–score

48. Checking for Normality
Construct a “normal probability plot” of the data. If the data are approximately normal, the points will fall approximately on a straight line.
Suppose the sample has mean X and standard deviation s. Then the normal probability plot plots:
X Axis: Actual value (and suppose its percentile is p)
Y Axis: The expected pth percentile from a normal distribution with mean X and standard deviation s (i.e., the “expected normal value”) _ _

50. Summary
1. From an open JMP data table, select Analyze > Distribution.
2. Select one or more continuous variables from Select Columns and click Y, Columns.
3. Click OK to generate a histogram and descriptive statistics.
4. Click on the red triangle for the variable and select Normal Quantile Plot.
5. If the data more-or-less follow a straight line (fat pen test), we can conclude that the data came from a normal distribution.
6. Select Continuous Fit > Normal from the lower red triangle.
7. In the resulting output, click on the red triangle for Fitted Normal and select Goodness of Fit.
8. A Prob< W value less than 0.05 indicates nonnormality.

51. Example with n = 100 weights
Visual (fat pencil) test: Looks good, conclude distribution is normal
Prob < W (p-value)test: value > .05, so conclude distribution is normal

52. Normal Plots of Residuals
N = 32 Normally Distributed Residuals
Bell Shape Straight Line

53. Normal Plots of Residuals: Patterns
Outliers on both
sides: “S” Shape
Investigate outliers
Skewed right: curving down
Take log of Y (quite common)
Skewed left: curving up
Rarely happens

54. Sampling Distributions

55. Parameter & Statistic Parameter Sample Statistic
Summary measure about population
Sample Statistic
Summary measure about sample
P in Population & Parameter
S in Sample & Statistic

56. Common Statistics & Parameters
Sample Statistic
Population Parameter
Mean 
Standard Deviation s 
Variance s2 2
Binomial Proportion p ^

57. Sampling Distribution
Theoretical probability distribution
Random variable is sample statistic
Sample mean, sample proportion, etc.
Results from drawing all possible samples of a fixed size
4. List of all possible [x, p(x)] pairs
Sampling distribution of the sample mean

58. Developing Sampling Distributions
Sony would like to estimate the average number of TVs per household in the U.S. using a sample of size 2!!
X = number of TVs in a household in the U.S.
Values of X: 1, 2, 3, 4
Assume we secretly know that TVs have a uniform distribution from 1 to 4 (nobody has zero TVs and nobody has more than 4)
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59. Population Characteristics
Summary Measures
Population Distribution
P(x)
Have students verify these numbers. x 1 2 3 4

60. All Possible Samples of Size n = 2
2nd Observation 1 2 3 4
1st
Obs
16 Sample Means
2nd Observation 1 2 3 4
1st
Obs
1,1
1,2
1,3
1,4
1.0
1.5
2.0
2.5
2,1
2,2
2,3
2,4
1.5
2.0
2.5
3.0
3,1
3,2
3,3
3,4
2.0
2.5
3.0
3.5
4,1
4,2
4,3
4,4
2.5
3.0
3.5
4.0
Sample with replacement

61. Sampling Distribution of All Sample Means
2nd Observation 1 2 3 4
1st
Obs
16 Sample Means
Sampling Distribution of the Sample Mean .0 .1 .2 .3
1.0
1.5
2.0
2.5
3.0
3.5
4.0
P(x) x
1.0
1.5
2.0
2.5
1.5
2.0
2.5
3.0
2.0
2.5
3.0
3.5
2.5
3.0
3.5
4.0

62. Summary Measures of All Sample Means
Sampling Distribution of Sample Mean
Population Parameters .0 .1 .2 .3
1.0
1.5
2.0
2.5
3.0
3.5
4.0
P(x) x
Have students verify these numbers.

63. Population Distribution of X Sampling Distribution of
Comparison
Population Distribution of X
Sampling Distribution of
P(x) .0 .1 .2 .3 1 2 3 4 .0 .1 .2 .3
1.0
1.5
2.0
2.5
3.0
3.5
4.0
P(x) x x

64. Standard Error of the Mean
1. Standard deviation of all possible sample means, x ● Measures scatter in all sample means, x
Less than population standard deviation
3. Shortcut formula:

65. Summary: Properties of the Sampling Distribution of x
Regardless of the sample size,
The mean of the sampling distribution equals the population mean
An estimator is a random variable used to estimate a population parameter (characteristic).
Unbiasedness
An estimator is unbiased if the mean of its sampling distribution is equal to the population parameter.
Efficiency
The efficiency of an unbiased estimator is measured by the variance of its sampling distribution.
If two estimators, with the same sample size, are both unbiased, then the one with the smaller variance has greater relative efficiency.
Consistency
An estimator is a consistent estimator of a population parameter if the larger the sample size, the more likely it is that the estimate will come close to the parameter.
The standard deviation of the sampling distribution equals
3. And what about the shape of the sampling distribution?

66. Central Limit Theorem X As sample size gets large enough ...
sampling distribution becomes bell shaped!!! X

67. JMP Demo of Sampling Distribution Sample Mean
Open the file “TVs in Population of HHs”
From JMP, open Dist_Sample_Means.jsl
Edit >> Run Script
For population shape use the pull down menu and select “My Data” and specify the TV population data table.
Choose sample size = 2, number of samples = 100, and animate = yes.
Press “Draw Samples”

68. Results will look something like this:

69. Population (Probability Dist’n)

70. Sample Size = 2

71. Sample Size = 4

72. Sample Size = 8

73. Sample Size = 16

74. Sample Size = 32

75. Sample Size = 1 (Population)

76. Sample Size= 2

77. Sample Size= 8

78. Sample Size = 16

79. Sample Size = 32

80. Summary: Sampling from Normal or Non-Normal Populations
Central Tendency
Dispersion
Sampling with replacement
Population Distribution s
= 10 m
= 50 X
Sampling Distribution
n = 4  = 5
n =30  = 1.8 m -
= 50 X X

81. Thinking Challenge
You’re an operations analyst for AT&T. Long-distance telephone calls are normally distribution with  = 8 min. and  = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes?
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82. Central Limit Theorem Example
The amount of soda in cans of a particular brand has a mean of 12 oz and a standard deviation of .2 oz. If you select random samples of 50 cans, what percentage of the sample means would be less than oz?
SODA

83. Conclusion Distinguished Between the Two Types of Random Variables
Described Discrete Probability Distributions
Described the Uniform and Normal Distributions
Explained Sampling Distributions
Solved Probability Problems Involving Sampling Distributions
As a result of this class, you will be able to...