This document summarizes key concepts from Chapter 5 of a statistics textbook, including discrete random variables and their probability distributions. Section 5.1 defines discrete random variables as variables that can take on countable values and introduces probability distributions and histograms to show the probabilities of different outcomes. Section 5.2 provides examples of defining random variables and constructing probability distributions. Section 5.3 covers the binomial distribution for experiments with two possible outcomes. Section 5.4 introduces the Poisson distribution for modeling rare events.

1. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 1
Chapter 5
Discrete Random
Variables
Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 1

2. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 2
Chapter 5
Discrete Random Variables

3. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 3
Section 5.1
Discrete Random Variables
and Probability Distributions

4. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 4
Definitions 5.1 & 5.2
Random Variable
A random variable is a quantitative variable whose value
depends on chance.
Discrete Random Variable
A discrete random variable is a random variable whose
possible values can be listed. In particular, a random
variable with only a finite number of possible values is a
discrete random variable.

5. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 5
{X = 4}
X = number of siblings of a randomly selected
student.
P(X=4)

6. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 6
Definition 5.3
Probability Distribution and Probability Histogram
Probability distribution: A listing of the possible values
and corresponding probabilities of a discrete random
variable, or a formula for the probabilities.
Probability histogram: A graph of the probability
distribution that displays the possible values of a discrete
random variable on the horizontal axis and the
probabilities of those values on the vertical axis. The
probability of each value is represented by a vertical bar
whose height equals the probability.

7. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 7
Table 5.2 & Figure 5.1
Probability distribution of the
random variable X, the number
of siblings of a randomly
selected student
Probability histogram for the random
variable X, the number of siblings of
a randomly selected student

8. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 8
Key Fact 5.1

9. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 9
Key Fact 5.2
Interpretation of a Probability Distribution
In a large number of independent observations of a random
variable X, the proportion of times each possible value occurs
will approximate the probability distribution of X; or,
equivalently, the proportion histogram will approximate the
probability histogram for X.

10. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 10
Figure 5.2
(a) Histogramof proportions for the numbers of heads obtained in three tosses of
a balanced dime for 1000 observations; (b) probability histogram for the number
of heads obtained in three tosses of a balanced dime

11. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 11
Section 5.3
The Binomial Distribution

12. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 12
Testing the effectiveness of a drug:
1. effective
2. not effective
Born of baby is BOY
1.YES
2. NO
Taste test for coca-cola:
1. first coca-cola
2. second coca-cola.

13. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 13
Definition 5.6
5! = 5x4x3x2x1

14. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 14
Definition 5.7
(a+b)n

15. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 15
Section 5.4
The Poisson Distribution

16. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 16
Formula 5.3

17. Copyright © 2017 Pearson Education, Ltd. Chapter 5, Slide 17
ƛ = 6.9
P(X≤2) = P(X=0)+P(X=1)+P(X=2) = 0.032
e(2.718) -6.9 (6.9)0
0!