36-225 - Handout for Lecture # 16
October 1, 2008
Special Discrete Random Variables - Review
I.
The Binomial Random Variable –
X
∼
Bin(
n, p
).
X
counts the number of successes out of
n
independent (Bernoulli) trials, each having a probability
of “success”
p
.
II.
The Geometric Random Variable –
X
∼
Geom(
p
).
Consider a series of independent (Bernoulli) trials, each having probability of “success”
p
.
X
is the number of trials until the
first
“success”.
Comment:
The geometric random variable is the only discrete random variable that has the
memoryless property:
P
(
X > a
+
b
|
X > a
) =
P
(
X > b
).
III.
The Negative Binomial Random Variable –
X
∼
NB(
r, p
).
Consider a series of independent (Bernoulli) trials, each having probability of “success”
p
.
X
is the number of trials until we get
r
“successes”.
Clearly, NB(1
, p
) is the same as Geom(
p
).
IV.
The Hypergeometric Random Variable –
X
∼
HG(
n, N, r
).
A sample of size
n
is chosen (without replacement) from a group of size
N
that has
r
“successes”
(and
N
-
r
“failures”).
X
is the number “successes” in the sample.
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- Spring '11
- finegold
- Statistics, Bernoulli, Binomial, Probability, Probability theory, Binomial distribution, Discrete probability distribution, Negative binomial distribution
-
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