p. 421
proof
.
p. 422
Summary for
X
~ Binomial(
n
,
p
)
Range:
Pmf:
Parameters:
n
∈
{1, 2, 3, …} and 0
≤
p
≤
1
Mean:
E
(
X
)=
np
Variance:
Var
(
X
)=
np
(1
p
)
X
=
{
0
,
1
,
2
,...,n
}
f
X
(
x
)=
¡
n
x
¢
p
x
(1
−
p
)
n
−
x
,
for
x
∈
X
• Geometric and Negative Binomial Distributions
Experiment: A basic experiment with sample space
0
is
repeated
infinite
times.
The sample space is
=
0
×
0
×
0
×
Assume
that events depending on different trials are
independent
For a given event
A
0
⊂
0
, we continue performing the trials
until
A
0
occurs exactly
r
times
Q
: What is the probability that we need to perform
k
trials?
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Example.
A company must hire 3 engineers.
Each interview results in a hire with probability 1/3
Q
: What is the probability that 10 interviews are required?
We need: (i) 2 hires on the first 9 interview (ii) Success on
the 10
th
interview
So, the probability is
μ
9
2
¶μ
1
3
¶
2
μ
2
3
¶
7
×
μ
1
3
¶
=
μ
9
2
1
3
¶
3
μ
2
3
¶
7
.
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 Fall '11
 ShaoWeiCheng
 Binomial, Poisson Distribution, Probability, Variance, Probability theory, Binomial distribution, Discrete probability distribution, Negative binomial distribution, Simeon Poisson

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