The mean of any discrete random variable is an average of the
possible outcomes, with each outcome weighted by its probability.
Mean of a Discrete Random Variable
Suppose that
X
is a discrete random variable whose probability
distribution is
Value:
x
1
x
2
x
3
…
Probability:
p
1
p
2
p
3
…
To find the
mean (expected value)
of
X
, multiply each possible value
by its probability, then add all the products:
𝜇
𝑥
= 𝐸
𝑋
= 𝑥
1
𝑝
1
+ 𝑥
2
𝑝
2
+ 𝑥
3
𝑝
3
+ ⋯
= 𝑥
𝑖
𝑝
𝑖

Example: Babies
’
Health at Birth
22
The probability distribution for
X
= Apgar scores is shown below:
a. Show that the probability distribution for
X
is legitimate.
b. Make a histogram of the probability distribution. Describe what you see.
c. Apgar scores of 7 or higher indicate a healthy baby. What is
P
(
X
≥
7)?
Value:
0
1
2
3
4
5
6
7
8
9
10
Probability:
0.001
0.006
0.007
0.008
0.012
0.020
0.038
0.099
0.319
0.437
0.053
(a) All probabilities are
between 0 and 1, and
they add up to 1. This
is a legitimate
probability distribution.
(b) The left-skewed shape of the distribution suggests a randomly selected
newborn will have an Apgar score at the high end of the scale. There is a small
chance of getting a baby with a score of 5 or lower.
(c)
P
(
X
≥ 7) = .908.
We
’
d have a 91%
chance of randomly
choosing a healthy
baby.

Example: Apgar Scores―What’
s
Typical?
23
Consider the random variable
X
= Apgar Score.
Compute the mean of the random variable
X
and interpret it in
context.
Value:
0
1
2
3
4
5
6
7
8
9
10
Probability:
0.001
0.006
0.007
0.008
0.012
0.020
0.038
0.099
0.319
0.437
0.053
𝜇
𝑥
= 𝐸
𝑋
= 𝑥
𝑖
𝑝
𝑖
=
0
0.001 +
1
0.006 +
2
0.007 + ⋯ + (10)(0.053)
= 8.128
The mean Apgar score of a randomly selected newborn is 8.128. This is the
long-term average Apgar score of many, many randomly chosen babies.
Note:
The expected value does not need to be a possible value of
X
or an
integer! It is a long-term average over many repetitions.

Statistical Estimation
24
Suppose we would like to estimate an unknown
μ
. We could select an
SRS and base our estimate on the sample mean. However, a different
SRS would probably yield a different sample mean.
This basic fact is called
sampling variability
: The value of a statistic
varies in repeated random sampling.
To make sense of sampling variability, we ask,
“
What would happen if we
took many samples?
”
Population
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample

The Law of Large Numbers
25
.
of
values
different
produce
would
samples
random
different
all,
After
?
of
estimate
accurate
an
be
can
How
x
μ
x
.
m
parameter
the
to
closer
and
closer
get
to
guaranteed
is
statistic
the
samples,
larger
and
larger
taking
on
keep
we
If
x
Draw independent observations at random from any population with
finite mean
μ
. The
law of large numbers
says that, as the number
of observations drawn increases, the sample mean of the observed
values gets closer and closer to the mean
μ
of the population.
Many people incorrectly believe the
law of small numbers
, which says that
we expect even short sequences of random events to show the kind of
average behavior that, in fact, appears only in the long run.