of \(X\)?We previously determined that the moment generating function of a binomial random variable is:for \(-\inftyt\infty\). How does it work? Well, here are the steps you usually follow:Lets do a simple example. If you have several distances and associated weights, you can calculate all the products and sum them and get what you call the first moment about the origin.
Related to the moment-generating function are a number of other transforms that are common in probability theory:
The moment generating function (mgf) is a function often used to characterize
the distribution of a random variable. They sum to 1. Let \(X_n \sim Bin(n,p_n)\) for all \(n \geq 1\), where \(np_n\) is a constant \(\lambda 0\) for all \(n\) (so \(p_n = \lambda/n\)).
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If random variable
X
{\displaystyle X}
has moment generating function
M
X
(
t
)
{\displaystyle M_{X}(t)}
, then
X
+
{\displaystyle \alpha X+\beta }
has moment generating function
X
+
(
t
)
=
e
t
M
X
(
t
)
{\displaystyle M_{\alpha X+\beta }(t)=e^{\beta t}M_{X}(\alpha t)}
If
S
n
=
i
=
1
n
a
i
X
i
{\displaystyle S_{n}=\sum _{i=1}^{n}a_{i}X_{i}}
, where the Xi are independent random variables and the ai are constants, then the probability density function for Sn is the convolution of the probability density functions of each of the Xi, and the moment-generating function for Sn is given by
For vector-valued random variables
X
{\displaystyle \mathbf {X} }
with real components, the moment-generating function is given by
where
t
{\displaystyle \mathbf {t} }
is a vector and
{\displaystyle \langle \cdot ,\cdot \rangle }
is the dot product. .