There are two questions that we avoided in our discussions of the
special discrete distributions:
(1)
How did we get from E(X) =
Σ
x*p(x)
and V(X) = E(x
2
)
[E(x)]
2
to the formulas specified for the means and variances
for the binomial, geometric, ….. distributions?
Note: Since x isn’t always bounded, it is difficult to use the
above formulas for E(x) and E(x
2
).
The answer to (1) is to use the moment generating function to get
the moments.
What are moments? The mean of a distribution is the 1
st
moment.
The 2
nd
moment is E(x
2
).
How do we find the moment generating function?
The moment generating function is denoted m
x
(t) and calculated as
E(e
tx
). It is unique for each distribution.
For discrete probability distributions, this means m
x
(t) =
Σ
e
tx
*f(x).
Evaluating these sums requires the application of special theorems
and sums.
How do we get the moments from the moment generating
function? Find
Binomial Distribution:
denote q=1p
m
x
(t) = E(e
tx
) =
Σ
e
tx
*f(x) =
= (q +pe
t
)
n
This requires use of the binomial theorem: (a+b)
n
=
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 '07
 MQLemons
 Binomial, Variance, Probability theory, Discrete probability distribution, moment generating function

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