1. Chapter 5 Discrete Random Variables and Probability Distributions
Instructor: Nahid Farnaz (Nhn)
North South University
Statistics for
Business and Economics 6th Edition
Chapter 5
Discrete Random Variables and Probability Distributions
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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. Question: Outcome of an experiment
For the experiment of flipping a coin twice, the random variable X is defined to be the number of tails and Y is defined to be the number of heads minus tails.
Construct a table to show the outcomes and the numerical value each random variable assigns to the outcome
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

4. Both variables X and Y involve counting.
OUTCOME
VALUE OF X
VALUE OF Y HH 2 HT 1 TH TT -2
The random variables here are discrete since they have a finite number of different values.
Both variables X and Y involve counting.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

5. Discrete Random Variables
Can only take on a countable number of values
Examples:
Roll a die twice
Let X be the number of times 4 comes up
(then X could be 0, 1, or 2 times)
Toss a coin 5 times.
Let X be the number of heads
(then X = 0, 1, 2, 3, 4, or 5)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

6. Examples of Discrete Random Variables
The number of defective items in a sample of 20 items from a large shipment
The number of customers arriving at a checkout counter in an hour
The number of errors detected in a corporation’s accounts
The number of claims on a medical insurance policy in a particular year
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

7. 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

8. Exercise 5.1
A store sells 0 to 12 computers per day. Is the daily computer sales a discrete or continuous random variable?
Answer: discrete random variable
Practice: Exercise 5.2, 5.3 and 5.4 (6th edition)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

9. 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.

10. Example 5.2: Rolling a die (Probability Function)
What is the probability function for the roll of a single six-headed balanced die?
Solution: Let the random variable X denote the number resulting from a single roll of a six-sided balanced die.
P(X=1) = P(X=2) = P(X=3)= P(X=4) =P(X=5) =P(X=6) =
The probability function is:
P(x) = P(X=x) = for x = 1, 2, 3,….,6
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

11. Probability Distribution Required Properties
0 ≤ P(x) ≤ 1 for any value of x
The individual probabilities sum to 1;
(The notation indicates summation over all possible x values)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

12. Cumulative Probability Function
The cumulative probability function, denoted F(x0), shows the probability that X is less than or equal to x0
In other words,
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

13. Exercise 5.14
American Air Travel has asked you to study flight delays during the week before Xmas. The random variable X is the number of flights delayed per hour.
What is the cumulative probability distribution?
What us the probability of five or more delayed flights?
What is the probability of three through seven (inclusive) delayed flights? X 1 2 3 4 5 6 7 8 9
P(x)
0.10
0.08
0.07
0.15
0.12

14. Exercise 5.13
The number of computers sold per day at Dan’s Computer works is defined by the following probability distribution:
What is the cumulative probability distribution?
P(3 ≤ x < 6)= ?
P(x > 3)= ?
P(x ≤ 4)= ?
P(2 < x ≤ 5) = ? X 1 2 3 4 5 6
P(x)
0.05
0.10
0.20
0.15
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

15. Properties of Discrete Random Variable
Expected Value (or mean) of a discrete
distribution (Weighted Average)
Example: Toss 2 coins,
x = # of heads,
compute expected value of x:
E(x) = (0 x .25) + (1 x .50) + (2 x .25)
= 1.0
x P(x)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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

17. Example 5.5: 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.

18. Example 5.5 (continued)
Find the mean and variance for completion time X
Find the mean, variance and standard deviation for total cost C.
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.

19. 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.

20. 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 are……….
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

21. 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.

22. Bernoulli 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.

23. 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.

24. 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.

25. Example: 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.

26. 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.

27. Binomial Characteristics
Examples
n = 5 P = 0.1
Mean
P(x) .6 .4 .2 x 1 2 3 4 5
n = 5 P = 0.5
P(x) .6 .4 .2 x 1 2 3 4 5
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

28. Example 5.7: 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.

29. 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.