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

32. Normal Distribution

33. 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

34. 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

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

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

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

38. Using Minitab for Normal Probabilities
Minitab only gives the cumulative probability P(X < UB) where “UB” is an upper bound (and LB is lower bound). UB
Three hints:
P(X < UB) (Minitab gives this directly)
P(LB < X < UB) = P(X < UB) - P(X < LB) (So use Minitab twice and compute the difference
P(X > LB) = 1 - P(X < UB) (So use Mintab once and then subtract from 1.0 using the complimentation law)

39. Using Minitab for Normal Probabilities
X follows N(100,) distribution; find P(X < 115)
Minitab menu sequence:
CALC >> Probability Distributions >> Normal
m=100 and s2=25 so s=5

40. 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.

41. 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 INVERSE of finding normal probabilities:
Before: Given an X value, find the cumulative probability
Now: Given a cumulative probability, find the X value
In Minitab, we use the “Inverse cumulative probability” option

42. Problem: Find the .35 percentile for a normal distribution with mean 20 and standard deviation 5.
Need to click the “Inverse” button!! Then put the percentile you need here.

43. 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?

44. Assessing Normality

45. 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

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

47. 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 pth percentile from an actual normal distribution with mean X and standard deviation s (i.e., the “expected normal value”) _ _

48. Checking for Normality
This is sometimes called a normal quantile-quantile plot

49. Checking for Normality
An absolutely equivalent plot can be obtained by re-labeling the Y axis using the percentiles rather than the normal values corresponding to the percentiles. This gives a normal probability plot and is what Minitab does:
Different
Label:
Same Plot!
If “P-value” is less than .05, conclude NOT normal

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

51. 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

52. Sampling Distributions

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

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

55. 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

56. Developing Sampling Distributions
Suppose There’s a Population ...
Population size, N = 4
Random variable, x
Values of x: 1, 2, 3, 4
Uniform distribution
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57. Population Characteristics
Summary Measures
Population Distribution
P(x) .3 .2
Have students verify these numbers. .1 .0 x 1 2 3 4

58. 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

59. 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

60. Summary Measures of All Sample Means
Have students verify these numbers.

61. Sampling Distribution
Comparison
Population
Sampling Distribution
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

62. 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. Formula (sampling with replacement)

63. 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?

64. Central Limit Theorem X As sample size gets large enough (n  30) ...
sampling distribution becomes almost normal. X

65. Population (Probability Dist’n)

66. Sample Size = 2

67. Sample Size = 4

68. Sample Size = 8

69. Sample Size = 16

70. Sample Size = 32

71. Sample Size = 1 (Population)

72. Sample Size= 2

73. Sample Size= 8

74. Sample Size = 16

75. Sample Size = 32

76. 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 X = 5
n =30 X = 1.8 m -
= 50 X X

77. 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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78. 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

79. 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...