Page 213 - Applied statistics and probability for engineers

Section 5-6/Moment-Generating Functions 191

riders can generate 6.6 watts per kilogram of body weight for power is computed as the fourth root of the mean of Y = X .
4
extended periods of time. Some meters calculate a normal- Determine the following:
ized power measurement to adjust for the physiological effort (a) Mean and standard deviation of X
required when the power output changes frequently. Let the (b) f y ( )
Y
random variable X denote the power output at a measure- (c) Mean and variance of Y
ment time and assume that X has a lognormal distribution (d) Fourth root of the mean of Y
with parameters θ = .5 2933 and ω = 0 00995. . The normalized (e) Compare [ (E X 4 )] to E X) and comment.
2
/
1 4
(
5-6 Moment-Generating Functions
Suppose that X is a random variable with mean μ. Throughout this book we have used the idea
of the expected value of the random variable X, and in fact E X( ) = μ. Now suppose that we
r
are interested in the expected value of a function of X, g X( ) = X . The expected value of this
function, or E g X[ ( )] = E X ), is called the rth moment about the origin of the random vari-
r
(
′
able X, which we will denote by μ r .
Deinition of Moments
about the Origin The rth moment about the origin of the random variable X is
⎧ ∑ x f x), X discrete
(
r
⎪
r
(
μ⎟ r = E X ) = ⎨ ∞ x (5-32)
⎪ ∫ x f x dx, X continuous
(
)
r
⎩ −∞
′
Notice that the irst moment about the origin is just the mean, that is, E X( ) = μ 1 = μ. Fur-
2
thermore, since the second moment about the origin is E X( ) = μ′ , we can write the variance
2
of a random variable in terms of origin moments as follows:
′
2
2
(
[
(
σ = E X ) − E X)] 2 = μ 2 − μ 2
The moments of a random variable can often be determined directly from the deinition in
Equation 5-32, but there is an alternative procedure that is frequently useful that makes use of
a special function.
Deinition of a
Moment-Generating The moment-generating function of the random variable X is the expected value of
Function e and is denoted by M t( ). That is,
tX
X
⎧ ∑ e f x ( ), X discrete
tx
⎪
⎨
M t ( ) = E e ( tX ) = ⎨ ∞ x (5-33)
X
⎪ ∫ e f x dx, X continuous
( )
tx
⎩ −∞
The moment-generating function M t( ) will exist only if the sum or integral in the above dei-
X
nition converges. If the moment-generating function of a random variable does exist, it can be
used to obtain all the origin moments of the random variable.
Let X be a random variable with moment-generating function M t( ). Then
X
r
X
μ⎟r = d M t ( ) t = 0 (5-34)
r
dt