Download 4.5 Probability generating functions

CHAPTER 4. DISCRETE RANDOM VARIABLES
4.5
48
Probability generating functions
We saw in the previous section that we could derive the mean and variance of a
Binomially distributed random variable but it was quite hard work. In this section
we introduce a method of finding the mean and variance of a discrete random
variable which is often useful.
We use the idea of a generating function. The particular function we employ is
called a probability generating function. This is defined as
X
GX (s) = E[sX ] =
sx P (X = x)
x
for any real s for which the sum converges. If we expand the sum we see that the
coefficient of sx is the probability that X = x. The quantity sP
is a dummy variable,
we could have used any letter we chose. Note that GX (1) = x P (X = x) = 1.
Now we can see why GX is useful if we differentiate it with respect to s
X
G′X (s) =
xsx−1 P (X = x)
x
and put s = 1
G′X (1) =
X
xP (X = x) = [X].
x
Similarly if we differentiate again with respect to s
X
G′′X (s) =
x(x − 1)sx−2P (X = x)
x
and put s = 1
G′′X (1) =
X
x(x − 1)P (X = x) = [X(X − 1)].
x
Now
V ar[X] = E[X 2 ] − (E[X])2
= E[X(X − 1)] + E[X] − (E[X])2
= G′′X (1) + G′X (1) − [G′X (1)]2 .
In general
dk GX (1)
E[X(X − 1)(X − 2)...(X − k + 1)] =
dsk
k
X (l)
where d G
is the kth derivative of ΠX (s) wrt s and evaluated at s = 1. Note
dsk
that
probability generating function is only going to be useful if we can simplify
P the
x
s
P
(X
= x).
x
Example 4.6
Discrete rv X has probability density function
1
a−x e− a
P (X = x) =
x!
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CHAPTER 4. DISCRETE RANDOM VARIABLES
1
1. Show that probability generating function Gx (s) for X is equal to e a (s−1) .
2. Using Gx (s) find E[X].
3. Using Gx (s) find V ar[X].
Solution
P
P∞ x a−x e− a1
1 P
x
1. Gx (sx ) = E[sx ] = ∞
= e− a ∞
x=0 s P (X = x) =
x=0 s
x=0
x!
1
s
1
e− a × e a = e a (s−1)
(sa−1 )x
x!
=
1
2. G′x (s) = a1 e a (s−1)
E[X] = G′x (1) =
1
a
1
3. G′′x (s) = a12 e a (s−1)
E[X(X − 1)] = G′′x (1) = a12
V ar[X] = E[X(X − 1)] + E[X] − (E[X])2 =
1
a2
+
1
a
−
1
a2
=
1
a
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