STAT 200
Chapter 5
Sampling Distributions
The binomial distribution for counts
(Section 5.1)
1. The Binomial Experiment
(a) The experiment consists of
n
identical trials.
The number of trials is fixed in
advance.
(b) The outcomes on each trial can be dichotomized into two complementary cate
gories: “success” and “failure”.
(c) The probability of the success event,
p
, is the same from trial to trial.
The
probability of the failure event is 1

p
.
(d) The
n
trials are independent of each other. In other words, the results of previous
trials do not affect the result of the current trial.
Under the conditions of a binomial experiment, the random variable
X
representing
the number/count of successes out of the
n
trials is a binomial random variable with
parameters
n
and
p
. We write
X
∼
Bin
(
n, p
).
2. For
X
∼
Bin
(
n, p
), the probability that
X
will take on a value of
k
is given by
P
(
X
=
k
) =
n
k
p
k
(1

p
)
n

k
,
k
= 0
,
1
,
2
,
· · ·
, n
where
n
k
=
n
!
k
!(
n

k
)!
is called the binomial coefficient (Another notation is
n
C
k
). It gives the total number
of ways (or combinations of successes and failures) of having
X
=
k
.
Note that
k
! =
k
×
(
k

1)
×
(
k

2)
×
...
×
2
×
1 and 1! = 1, 0! = 1.
3. Mean (expected value), Variance and SD for a Binomial Random Variable
X
•
Mean
μ
X
:
E
(
X
) =
n
k
=0
k
×
P
(
X
=
k
) =
n
k
=0
k
×
n
k
p
k
(1

p
)
n

k
=
np
1
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•
Variance
σ
2
X
:
V
(
X
)
=
n
k
=0
(
k

np
)
2
×
P
(
X
=
k
)
=
n
k
=0
(
k

np
)
2
×
n
k
p
k
(1

p
)
n

k
=
np
(1

p
)
•
Standard deviation
σ
X
=
SD
(
X
) =
np
(1

p
)
•
Interpretation of
E
(
X
):
the average number of successes if you are to repeat the binomial experiment
(each with
n
trials) many times.
•
Interpretation of
V
(
X
):
a measure of the variability of the numbers of successes if the binomial experiment
(each with
n
trials) is repeated many times.
Sampling distribution
(Section 3.3)
•
Population vs. sample, parameter vs. statistic
The complete collection of individuals that we want to study about is called the
population
.
A census provides a means to obtain complete and accurate informa
tion about a population of interest. However, it can be very costly and time inefficient
to carry out. Sometimes a census is impossible.
It is more practical to study a sample
, i.e., a subset of individuals selected from a
population.
A sample can provide reliable information about a population as long
as the sample is representative of the population. A systematically nonrepresentative
sample is said to be biased
.
By randomly selecting individuals from the population, the sample produced tend
to have characteristics comparable to the population. The chance of obtaining a non
representative sample is thus minimized.
A parameter
refers to a numerical summary of a population.
The counterpart of a
sample is a statistic
. The value of a parameter is fixed yet unknown in practice. We
use statistics to estimate population parameters. Due to sampling variability (varia
tion from sample to sample), a statistic takes on different values for different samples.
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 Spring '08
 KARIM
 Binomial, Normal Distribution, Standard Deviation, Probability theory

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