1. Chapter 4 Discrete Random Variables and Probability Distributions
Statistics for
Business and Economics 8th Edition
Chapter 4
Discrete Random Variables and Probability Distributions

2. Introduction to Probability Distributions
Random Variable: a variable that takes on numerical values determined by the outcome of a random experiment.
Random
Variables
Ch. 5
Discrete
Random Variable
Continuous
Random Variable
Ch. 6
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

3. RANDOM VARIABLES Discrete Random Variable Continuous Random Variable
Table 5.1 Frequency and Relative Frequency Distribution of the Number of Vehicles Owned by Families

4. RANDOM VARIABLES Definition
A random variable is a variable whose value is determined by the outcome of a random experiment.
A random variable that assumes countable values is called a discrete random variable.

5. Examples of discrete random variables
The number of cars sold at a dealership during a given month
The number of houses in a certain block
The number of fish caught on a fishing trip
The number of complaints received at the office of an airline on a given day
The number of customers who visit a bank during any given hour
The number of heads obtained in three tosses of a coin
A store sells 0 to 12 computers per day. Is the daily computer sales a discrete or continuous random variable?
Answer: discrete random variable

6. Continuous Random Variable
Definition
A random variable that can assume any value contained in one or more intervals is called a continuous random variable.
A continuous random variable can take any value in an interval
We can not assign probabilities to specific values for continuous random variables (for e.g. the probability that temperature will be exactly 77 degrees is 0)
But we can determine ranges and attach probabilities for the range
Examples of continuous random variables:
The height of a person
The time taken to commute from home to work
The weight of a fish
The price of a house

7. Discrete Probability Distribution
The probability that discrete random variable X takes specific value is denoted by P(X = x).
Probability Distribution Function, P(x):
Expresses the probability that X takes the value x, as a function of x.
That is,
P(x) = P(X = x) for all values of x
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

8. PROBABLITY DISTRIBUTION OF A DISCRETE RANDOM VARIABLE
Definition
The probability distribution of a discrete random variable lists all the possible values that the random variable can assume and their corresponding probabilities.
Example 5-1
Recall the frequency and relative frequency distributions of the number of vehicles owned by families given in Table 5.1. That table is reproduced below as Table 5.2. Let x be the number of vehicles owned by a randomly selected family. Write the probability distribution of x.

9. Table 5-2 Frequency and Relative Frequency Distributions of the Number of Vehicles Owned by Families
Example 5-1: Solution

10. Figure 5.1 Graphical presentation of the probability distribution of Table 5.3.

11. Two Characteristics of a Probability Distribution
The probability distribution of a discrete random variable possesses the following two characteristics.
0 ≤ P (x) ≤ 1 for each value of x
ΣP (x) = 1.

12. Example 5-2
Each of the following tables lists certain values of x and their probabilities. Determine whether or not each table represents a valid probability distribution.
No, since the sum of all probabilities is not equal to 1.0.
Yes
No since one of the probabilities is negative.

13. Example 5-3
The following table lists the probability distribution of the number of breakdowns per week for a machine based on past data.
Present this probability distribution graphically.
Find the probability that the number of breakdowns for this machine during a given week is
exactly 2 ii. 0 to 2
more than 1 iv. at most 1

14. Example 5-3: Solution
Let x denote the number of breakdowns for this machine during a given week. Table 5.4 lists the probability distribution of x.

15. Example 5-3: Solution (b) Using Table 5.4,
i. P(exactly 2 breakdowns) = P(x = 2) = .35
ii. P(0 to 2 breakdowns) = P(0 ≤ x ≤ 2) = P(x = 0) + P(x = 1) + P(x = 2) = = .70
iii. P(more then 1 breakdown) = P(x > 1) = P(x = 2) + P(x = 3)
= = .65
iv. P(at most one breakdown) = P(x ≤ 1) = P(x = 0) + P(x = 1)
= = .35

16. Example 5-4
According to a survey, 60% of all students at a large university suffer from math anxiety. Two students are randomly selected from this university.
Let x denote the number of students in this sample who suffer from math anxiety. Develop the probability distribution of x.
Let us define the following two events: N = the student selected does not suffer from math anxiety M = the student selected suffers from math anxiety

17. Example 5-4: Solution Figure 5. 3 Tree diagram
Example 5-4: Solution Figure 5.3 Tree diagram. (Think of flipping a coin twice. What will be the outcomes? HH, HT, TH, TT)

18. Example 5-4: Solution
Table 5.5 Probability Distribution of the Number of Students with Math Anxiety in a Sample of Two Students
Let us define the following two events: N = the student selected does not suffer from math anxiety M = the student selected suffers from math anxiety P(x = 0) = P(NN) = .16 P(x = 1) = P(NM or MN) = P(NM) + P(MN)
= = .48 P(x = 2) = P(MM) = .36

19. MEAN OF A DISCRETE RANDOM VARIABLE
The mean of a discrete variable x is the value that is expected to occur per repetition, on average, if an experiment is repeated a large number of times. It is denoted by µ and calculated as
µ = Σ x P(x)
The mean of a discrete random variable x is also called its expected value and is denoted by E(x); that is,
E(x) = Σ x P(x)

20. Example 5-5
Recall Example 5-3 of Section 5-2. The probability distribution Table 5.4 from that example is reproduced on the next slide. In this table, x represents the number of breakdowns for a machine during a given week, and P(x) is the probability of the corresponding value of x.
Find the mean number of breakdown per week for this machine.

21. Example 5-5: Solution Table 5
Example 5-5: Solution Table 5.6 Calculating the Mean for the Probability Distribution of Breakdowns
The mean is µ = Σx P(x) = 1.80

22. STANDARD DEVIATION OF A DISCRETE RANDOM VARIABLE
The standard deviation of a discrete random variable x measures the spread of its probability distribution and is computed as

23. Variance and Standard Deviation
Variance of a discrete random variable X
Standard Deviation of a discrete random variable X
Or,
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

24. Example 5-6
Baier’s Electronics manufactures computer parts that are supplied to many computer companies. Despite the fact that two quality control inspectors at Baier’s Electronics check every part for defects before it is shipped to another company, a few defective parts do pass through these inspections undetected. Let x denote the number of defective computer parts in a shipment of 400. The following table gives the probability distribution of x.

25. Solution: Example 5-6
Compute the standard deviation of x.

26. Example 5.7: Total project cost
A contractor is interested in the total cost of a project. He estimates that materials will cost $25,000 and labor will be $900 per day. If the project takes X days to complete, the total labor cost will be 900X dollars, and the total cost will be:
C=25, X
The contractor forms the following probabilities of likely completion times:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

27. Example 5.7 Linear Functions of Random Variables
Find the mean and variance for completion time X
Find the mean, variance and standard deviation for total cost C.
Linear Functions of Random Variables
X (days) 10 11 12 13 14
P(X)
0.1
0.3
0.2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

28. Special Linear Functions of Random Variables
Let a and b be any constants. a)
i.e., if a random variable always takes the value a,
it will have mean a and variance 0 b)
i.e., the expected value of b·X is b·E(x)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

29. THE BINOMIAL PROBABILITY DISTRIBUTION
Topics:
The Binomial Experiment
The Binomial Probability Distribution and Binomial Formula
Using the Table of Binomial Probabilities
Probability of Success and the Shape of the Binomial Distribution
Mean and Standard Deviation of the Binomial Distribution
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

30. Binomial Distribution
The binomial random variable is a discrete random variable that is defined when the conditions of a binomial experiment are satisfied.
The four conditions of binomial experiment are……….
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

31. Conditions of Binomial Probability Distribution
A fixed number of observations, n
e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse
Each observation has only two possible outcomes
e.g., head or tail in each toss of a coin; defective or not defective light bulb
Generally called “success” and “failure”
Probability of success is P , probability of failure is 1 – P
Constant probability for each observation
e.g., Probability of getting a tail is the same each time we toss the coin
Observations are independent
The outcome of one observation does not affect the outcome of the other
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

32. The Binomial Experiment
Conditions of a Binomial Experiment
A binomial experiment must satisfy the
following four conditions.
There are n identical trials.
Each trail has only two possible outcomes.
The probabilities of the two outcomes remain constant.
The trials are independent.
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

33. Example 5-16: Consider the experiment consisting of 10 tosses of a coin. Determine whether or not it is a binomial experiment.
Solution:
1. There are a total of 10 trials (tosses), and they are all identical. Here, n=10.
2. Each trial (toss) has only two possible outcomes: a head and a tail.
3. The probability of obtaining a head (success) is ½ and that of a tail (a failure) is ½ for any toss. That is,
p = P(H) = ½ and q = P(T) = ½
4. The trials (tosses) are independent.
Consequently, the experiment consisting of 10 tosses is a binomial experiment.

34. Example 5-17
Five percent of all DVD players manufactured by a large electronics company are defective. Three DVD players are randomly selected from the production line of this company. The selected DVD players are inspected to determine whether each of them is defective or good. Is this experiment a binomial experiment?
Solution:
1. This example consists of three identical trials.
2. Each trial has two outcomes: defective or good.
3. The probability p that a DVD player is defective is The probability q that a DVD player is good is .95.
4. Each trial (DVD player) is independent.
Because all four conditions of a binomial experiment are satisfied, this is an example of a binomial experiment.

35. Bernoulli Distribution (Basic building block of Binomial Distribution)
Consider only two outcomes: “success” or “failure”
Let P denote the probability of success
Let 1 – P be the probability of failure
Define random variable X:
x = 1 if success, x = 0 if failure
Then the Bernoulli probability function is
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

36. Bernoulli Distribution Mean and Variance
The mean is µ = P
The variance is σ2 = P(1 – P)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

37. The Binomial Probability Distribution and Binomial Formula
For a binomial experiment, the probability of exactly x successes in n trials is given by the binomial formula
where
n = total number of trials
p = probability of success
q = 1 – p = probability of failure
x = number of successes in n trials
n - x = number of failures in n trials
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

38. Binomial Distribution Formula! X n - X
P(x) = P
(1- P)
x ! ( n - x ) !
P(x) = probability of x successes in n trials,
with probability of success P on each trial
x = number of ‘successes’ in sample,
(x = 0, 1, 2, ..., n)
n = sample size (number of trials
or observations)
P = probability of “success”
Example: Flip a coin four times, let x = # heads:
n = 4
P = 0.5
1 - P = ( ) = 0.5
x = 0, 1, 2, 3, 4
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

39. Example 5-18: Calculating a Binomial Probability
What is the probability of one success in five observations if the probability of success is 0.1?
x = 1, n = 5, and P = 0.1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

40. Binomial Distribution Mean and Variance
Variance and Standard Deviation
Where n = sample size
P = probability of success
(1 – P) = probability of failure
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

41. Probability of Success and the Shape of the Binomial Distribution
The binomial probability distribution is symmetric if p = .50.
The binomial probability distribution is skewed to the right if p is less than .50.
The binomial probability distribution is skewed to the left if p is greater than .50.
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

42. Table 5.13 Probability Distribution of x for n = 4 and p= .50
The binomial probability distribution is symmetric if p = .50.

43. Table 5.14 Probability Distribution of x for n = 4 and p= .30
2. The binomial probability distribution is skewed to the right if p is less than .50.

44. Table 5.15 Probability Distribution of x for n = 4 and p= .80
The binomial probability distribution is skewed to the left if p is greater than .50.

45. Mean and Standard Deviation of the Binomial Distribution
The mean and standard deviation of a binomial distribution are, respectively,
where n is the total number of trails, p is the probability of success, and q is the probability of failure.
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

46. Binomial Distribution Mean and Variance
Variance and Standard Deviation
Where n = sample size
P = probability of success
(1 – P) = probability of failure
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

47. Example 5-22: Solution: n = 60 p = .56, and q = .44
According to a Harris Interactive survey conducted for World Vision and released in February 2009, 56% of teens in the United States volunteer time for charitable causes. Assume that this result is true for the current population of U.S. teens. A sample of 60 teens is selected. Let x be the number of teens in this sample who volunteer time for charitable causes. Find the mean and standard deviation of the probability distribution of x.
Solution:
n = p = .56, and q = .44
Using the formulas for the mean and standard deviation of the binomial distribution,

48. Example 5.19: Binomial Calculations
An insurance broker has five contacts and she believed that for each contact the probability of making a sale is 0.40.
Find the probability that she makes at most one sale
Find the probability that she makes between two and four sales (inclusive)
*Solve this using formula and Table 3 (Appendix)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

49. Using Binomial Tables Examples:…
p=.20
p=.25
p=.30
p=.35
p=.40
p=.45
p=.50 10 1 2 3 4 5 6 7 8 9
0.1074
0.2684
0.3020
0.2013
0.0881
0.0264
0.0055
0.0008
0.0001
0.0000
0.0563
0.1877
0.2816
0.2503
0.1460
0.0584
0.0162
0.0031
0.0004
0.0282
0.1211
0.2335
0.2668
0.2001
0.1029
0.0368
0.0090
0.0014
0.0135
0.0725
0.1757
0.2522
0.2377
0.1536
0.0689
0.0212
0.0043
0.0005
0.0060
0.0403
0.1209
0.2150
0.2508
0.2007
0.1115
0.0425
0.0106
0.0016
0.0025
0.0207
0.0763
0.1665
0.2384
0.2340
0.1596
0.0746
0.0229
0.0042
0.0003
0.0010
0.0098
0.0439
0.1172
0.2051
0.2461
Using Table 3 in the Appendix.
Examples:
n = 10, x = 3, P = 0.35: P(x = 3|n =10, p = 0.35) = .2522
n = 10, x = 8, P = 0.45: P(x = 8|n =10, p = 0.45) = .0229
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.