This chapter discusses discrete random variables and their probability distributions. It introduces the concept of a random variable and defines discrete and continuous random variables. It then covers the probability distributions for discrete random variables including the binomial, Poisson, and hypergeometric distributions. It defines key terms like expected value and variance and provides examples of calculating probabilities using these distributions.

1. Chapter 4: Discrete Random Variables
Statistics

2. McClave, Statistics, 11th ed. Chapter 4: Discrete
Random Variables
2
Where We’ve Been
 Using probability to make inferences
about populations
 Measuring the reliability of the
inferences

3. McClave, Statistics, 11th ed. Chapter 4: Discrete
Random Variables
3
Where We’re Going
 Develop the notion of a random
variable
 Numerical data and discrete random
variables
 Discrete random variables and their
probabilities

4. 4.1: Two Types of Random
Variables
 A random variable is a variable hat
assumes numerical values associated
with the random outcome of an
experiment, where one (and only one)
numerical value is assigned to each
sample point.
4McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

5. 4.1: Two Types of Random
Variables
 A discrete random variable can assume a
countable number of values.
 Number of steps to the top of the Eiffel Tower*
 A continuous random variable can
assume any value along a given interval of
a number line.
 The time a tourist stays at the top
once s/he gets there
*Believe it or not, the answer ranges from 1,652 to 1,789. See Great Buildings
5McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

6. 4.1: Two Types of Random
Variables
 Discrete random variables
 Number of sales
 Number of calls
 Shares of stock
 People in line
 Mistakes per page
 Continuous random
variables
 Length
 Depth
 Volume
 Time
 Weight
6McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

7. 4.2: Probability Distributions
for Discrete Random Variables
 The probability distribution of a
discrete random variable is a graph,
table or formula that specifies the
probability associated with each
possible outcome the random variable
can assume.
 p(x) ≥ 0 for all values of x
 Σp(x) = 1
7McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

8. 4.2: Probability Distributions
for Discrete Random Variables
 Say a random variable
x follows this pattern:
p(x) = (.3)(.7)x-1
for x > 0.
 This table gives the
probabilities (rounded
to two digits) for x
between 1 and 10.
x P(x)
1 .30
2 .21
3 .15
4 .11
5 .07
6 .05
7 .04
8 .02
9 .02
10 .01
8McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

9. 4.3: Expected Values of
Discrete Random Variables
 The mean, or expected value, of a
discrete random variable is
( ) ( ).E x xp xµ = = ∑
9McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

10. 4.3: Expected Values of
Discrete Random Variables
 The variance of a discrete random
variable x is
 The standard deviation of a discrete
random variable x is
2 2 2
[( ) ] ( ) ( ).E x x p xσ µ µ= − = −∑
2 2 2
[( ) ] ( ) ( ).E x x p xσ µ µ= − = −∑
10McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

11. )33(
)22(
)(
σµσµ
σµσµ
σµσµ
+<<−
+<<−
+<<−
xP
xP
xP
Chebyshev’s Rule Empirical Rule
≥ 0 ≅ .68
≥ .75 ≅ .95
≥ .89 ≅ 1.00
11McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
4.3: Expected Values of
Discrete Random Variables

12. 4.3: Expected Values of
Discrete Random Variables
12McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
 In a roulette wheel in a U.S. casino, a $1 bet on
“even” wins $1 if the ball falls on an even number
(same for “odd,” or “red,” or “black”).
 The odds of winning this bet are 47.37%
9986.
0526.5263.1$4737.1$
5263.)1$(
4737.)1$(
=
−=⋅−⋅+=
=
=
σ
µ
loseP
winP
On average, bettors lose about a nickel for each dollar they put down on a bet like this.
(These are the best bets for patrons.)

13. 4.4: The Binomial Distribution
 A Binomial Random Variable
 n identical trials
 Two outcomes: Success or Failure
 P(S) = p; P(F) = q = 1 – p
 Trials are independent
 x is the number of Successes in n trials
13McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

14. 4.4: The Binomial Distribution
 A Binomial Random
Variable
 n identical trials
 Two outcomes: Success
or Failure
 P(S) = p; P(F) = q = 1 – p
 Trials are independent
 x is the number of S’s in n
trials
Flip a coin 3 times
Outcomes are Heads or Tails
P(H) = .5; P(F) = 1-.5 = .5
A head on flip i doesn’t
change P(H) of flip i + 1
14McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

15. 4.4: The Binomial Distribution
Results of 3 flips Probability Combined Summary
HHH (p)(p)(p) p3
(1)p3
q0
HHT (p)(p)(q) p2
q
HTH (p)(q)(p) p2
q (3)p2
q1
THH (q)(p)(p) p2
q
HTT (p)(q)(q) pq2
THT (q)(p)(q) pq2
(3)p1
q2
TTH (q)(q)(p) pq2
TTT (q)(q)(q) q3
(1)p0
q3
15McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

16. 4.4: The Binomial Distribution
 The Binomial Probability Distribution
 p = P(S) on a single trial
 q = 1 – p
 n = number of trials
 x = number of successes
xnx
qp
x
n
xP −






=)(
16McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

17. 4.4: The Binomial Distribution
 The Binomial Probability Distribution
xnx
qp
x
n
xP −






=)(
17McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

18.  Say 40% of the
class is female.
 What is the
probability that 6
of the first 10
students walking
in will be female?
4.4: The Binomial Distribution
1115.
)1296)(.004096(.210
)6)(.4(.
6
10
)(
6106
=
=






=






=
−
−xnx
qp
x
n
xP
18McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

19. 4.4: The Binomial Distribution
Mean
Variance
Standard Deviation
 A Binomial Random Variable has
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
19
2
np
npq
npq
µ
σ
σ
=
=
=

20. 4.4: The Binomial Distribution
16250
2505.5.1000
5005.1000
2
≅==
=⋅⋅==
=⋅==
npq
npq
np
σ
σ
µ
 For 1,000 coin flips,
The actual probability of getting exactly 500 heads out of 1000 flips is
just over 2.5%, but the probability of getting between 484 and 516 heads
(that is, within one standard deviation of the mean) is about 68%.
20McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

21. 4.5: The Poisson Distribution
 Evaluates the probability of a (usually
small) number of occurrences out of many
opportunities in a …
 Period of time
 Area
 Volume
 Weight
 Distance
 Other units of measurement
21McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

22. 4.5: The Poisson Distribution
!
)(
x
e
xP
x λ
λ −
=
  = mean number of occurrences in the
given unit of time, area, volume, etc.
 e = 2.71828….
 µ = 
 2
= 
22McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

23. 4.5: The Poisson Distribution
1008.
!5
3
!
)5(
35
====
−−
e
x
e
xP
x λ
λ
 Say in a given stream there are an average
of 3 striped trout per 100 yards. What is the
probability of seeing 5 striped trout in the
next 100 yards, assuming a Poisson
distribution?
23McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

24. 4.5: The Poisson Distribution
0141.
!5
5.1
!
)5(
5.15
====
−−
e
x
e
xP
x λ
λ
 How about in the next 50 yards, assuming a
Poisson distribution?
 Since the distance is only half as long,  is only
half as large.
24McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables

25. 4.6: The Hypergeometric
Distribution
 In the binomial situation, each trial was
independent.
 Drawing cards from a deck and replacing
the drawn card each time
 If the card is not replaced, each trial
depends on the previous trial(s).
 The hypergeometric distribution can be
used in this case.
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
25

26. 4.6: The Hypergeometric
Distribution
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
26
 Randomly draw n elements from a set
of N elements, without replacement.
Assume there are r successes and N-r
failures in the N elements.
 The hypergeometric random variable
is the number of successes, x, drawn
from the r available in the n selections.

27. 4.6: The Hypergeometric
Distribution
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
27












−
−






=
n
N
xn
rN
x
r
xP )(
where
N = the total number of elements
r = number of successes in the N elements
n = number of elements drawn
X = the number of successes in the n elements

28. 4.6: The Hypergeometric
Distribution
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
28












−
−






=
n
N
xn
rN
x
r
xP )(
)1(
)()(
2
2
−
−−
=
=
NN
nNnrNr
N
nr
σ
µ

29. 4.6: The Hypergeometric
Distribution
44.22.2)2()2()2or2(
22.
45
)1)(10(
2
10
22
510
2
5
)2()2(
=×==+====
==












−
−






====
FPMPFMP
FPMP
 Suppose a customer at a pet store wants to buy two hamsters
for his daughter, but he wants two males or two females (i.e.,
he wants only two hamsters in a few months)
 If there are ten hamsters, five male and five female, what is the
probability of drawing two of the same sex? (With hamsters, it’s
virtually a random selection.)
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
29