Statistics
Chapter 4: Discrete Random Variables
Where We’ve Been


Using probability to make inferences
about populations.
Measuring the reliability of the
inferences.
McClave, Statistics, 11th ed. Chapter 4: Discrete
Random Variables
2
Where We’re Going



Develop the notion of a random
variable.
Numerical data and discrete random
variables.
Discrete random variables and their
probabilities.
McClave, Statistics, 11th ed. Chapter 4: Discrete
Random Variables
3
4.1: Two Types of Random
Variables

A random variable is a variable that
takes on numerical or categorical
values associated with the random
outcome of an experiment, where one
(and only one) numerical or categorical
value is assigned to each sample point.
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
4
4.1: Two Types of Random
Variables

A discrete random variable can take on a
countable number of values.


Number of steps to the top of the Eiffel Tower*
A continuous random variable can take
on 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.
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
5
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
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
6
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.


0 ≤ p(x) ≤ 1 for all values of x
 p(x) = 1
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
7
4.2: Probability Distributions
for Discrete Random Variables

Say a random variable
x follows this pattern:
p(x) = (0.3)(0.7)x-1
for x > 0.

This table gives the
probabilities (rounded
to two decimals) for x
between 1 and 10.
x
p(x)
1
0.30
2
0.21
3
0.15
4
0.11
5
0.07
6
0.05
7
0.04
8
0.02
9
0.02
10
0.01
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
8
4.3: Expected Values of
Discrete Random Variables

The mean, or expected value, of a
discrete random variable is
  E( X )   xp( x).
E(X) is read as
“expected value of X” or “mean of X”
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
9
4.3: Expected Values of
Discrete Random Variables

The variance of a discrete random
variable X is
 2  E ( X   ) 2    ( x   ) 2 p( x) .

The standard deviation of a discrete
random variable X is
   2  E ( X   )2  
2

(
x


)
p( x)  .

McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
10
4.3: Expected Values of
Discrete Random Variables
P(     X     )
P (   2  X    2 )
P (   3  X    3 )
Chebyshev’s Rule
Empirical Rule
≥0
 .68
≥ .75
 .95
≥ .89
 1.00
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
11
4.3: Expected Values of
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%
P( win $1)  0.4737
P(lose $1)  0.5263
  1 0.4737  (1)  0.5263  0.0526
 2  E ( X   ) 2   (1  .0526) 2 (0.4737)  (1  .0526) 2 (0.5263)
 0.5248  0.4724  0.9972
  0.9972  0.9986
On average, bettors lose about five cents for each dollar they put down on a bet like this.
(These are the best bets for patrons.)
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
12
4.4: The Binomial Distribution

A Binomial Random Variable





n identical trials
Two outcomes per trial: Success or Failure
P(S) = p; P(F) = q = 1 – p
Trials are independent
x is the number of Successes in n trials
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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 S’s in n
trials
Flip a coin 3 times
Outcomes are Heads (S) or
Tails (F)
P(H) = 0.5; P(T) = 1-0.5 =0.5
Result on a flip doesn’t affect
the outcomes of other flips
x heads in 3 coin flips
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
14
4.4: The Binomial Distribution
Results of 3 flips
Probability
Combined
Summary
HHH
(p)(p)(p)
p3
(1)p3q0
HHT
(p)(p)(q)
p2q
HTH
(p)(q)(p)
p2q
THH
(q)(p)(p)
p2q
HTT
(p)(q)(q)
pq2
THT
(q)(p)(q)
pq2
TTH
(q)(q)(p)
pq2
TTT
(q)(q)(q)
q3
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
(3)p2q1
(3)p1q2
(1)p0q3
15
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
 n  x n x
P( x)    p q
 x
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
16
4.4: The Binomial Distribution

The Binomial Probability Distribution
 n  x n x
P( x)    p q
 x
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
17
4.4: The Binomial Distribution


Say 40% of the
light bulbs in a
production line are
defective.
What is the
probability that 6 of
the 10 randomly
selected bulbs are
defective?
 n  x n x
P( x)    p q
 x
 10 
    0.46  0.610 6 
6
 210(0.004096)(0.1296)
 0.1115
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
18
4.4: The Binomial Distribution

A Binomial Random Variable has
Mean:
  E ( X )  np
Variance: 
Standard Deviation: 
2
 Var ( X )  npq
 SD( X )  npq
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
19
4.4: The Binomial Distribution

For 1,000 coin flips,
  np  1000 0.5  500
  npq  1000 0.5  0.5  250
2
  npq  250  16
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%.
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
20