Foundations of Data Science

Generating functions are useful for calculating the mean and standard deviation of a
sequence. Let z be an integral valued random variable where p i is the probability that
∑
z equals i. The expected value of z is given by m = ∞ ∑
ip i . Let p(x) = ∞ p i x i be the
generating function for the sequence p 1 , p 2 , . . .. The generating function for the sequence
p 1 , 2p 2 , 3p 3 , . . . is
x d
∞
dx p(x) = ∑
ip i x i .
Thus, the expected value of the random variable z is m = xp ′ (x)| x=1 = p ′ (1). If p was not
a probability function, its average value would be p′ (1)
since we would need to normalize
p(1)
the area under p to one.
i=0
i=0
i=0
The second moment of z, is E(z 2 ) − E 2 (z) and can be obtained as follows.
x 2 d ∣ ∣∣∣x=1
∞∑
dx p(x) = i(i − 1)x i p(x)
∣
i=0
x=1
∞∑
= i 2 x i ∞∑
p(x)
− ix i p(x) ∣
∣
x=1
i=0
= E(z 2 ) − E(z).
Thus, σ 2 = E(z 2 ) − E 2 (z) = E(z 2 ) − E(z) + E(z) − E 2 (z) = p ” (1) + p ′ (1) − ( p ′ (1) ) 2
.
12.8.2 The Exponential Generating Function and the Moment Generating
Function
Besides the ordinary generating function there are a number of other types of generating
functions. One of these is the exponential generating function. Given a sequence
∑
a 0 , a 1 , . . . , the associated exponential generating function is g(x) = ∞ x
a i
i . i!
Moment generating functions
The k th moment of a random variable x around the point b is given by E((x − b) k ).
Usually the word moment is used to denote the moment around the value 0 or around
the mean. In the following, we use moment to mean the moment about the origin.
i=0
∣
x=1
The moment generating function of a random variable x is defined by
i=0
Ψ(t) = E(e tx ) =
∫ ∞
e tx p(x)dx
−∞
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