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Moments and Moment
Generating Function
Moments
Definition: For each n, the nth moment of X is µn = E(X n ) and
the nth central moment of X is κn = E(X − µ)n where
µ = E(X ).
The moments can be used to measure some aspects of a
distribution. For example, we can measure degree of
asymmetric of a distribution by coefficient of skewness:
γ1 =
κ3
σ3
and for symmetric densities (pmfs), we can measure
peakedness by coefficient of kurtosis γ2 =
σ 2 = Var(X ).
κ4
σ4
− 3, where
Left skew and right skew
Examples: Skewness
0.4
0.3
p=0.1
p=0.8
0.0
0.1
0.1
0.2
0.3
probability
0.4
λ=2
λ=5
0.0
density
Binomial Distribution(n=10)
0.2
0.5
Exponential Distribution
0
2
4
6
x
8
10
0
2
4
6
x
8
10
Kurtosis
Kurtosis γ2 =
κ4
σ4
− 3 is compared with the kurtosis of the
standard normal distribution, which has kurtosis 0.
Moment generating function
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Let X be a random variable with CDF FX . The moment
generating function (MGF) of X , denoted by MX (t) is
MX (t) = E(etX ) provided that the expectation exists for t in
some neighborhood of 0. That is for all −h < t < h, E(etX )
exists.
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For continuous random variables with density function
R
fX (x), MX (t) = etx fX (x)dx.
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For discrete random variables with probability mass
P
function pX (x), MX (t) = x etx pX (x).
Calculate moments using MGF
If X has MGF MX (t), then
E(X n ) =
dn
MX (t)|t=0 .
dt n
That is, the n-th moment is equal to the n-th derivatives of
MX (t) evaluated at t = 0.
Examples
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(Discrete case) If X has Bernoulli(p), what is the moment
generating function of X ?
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(Continuous case) If X has Exponential(λ), what is the
moment generating function of X ?
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(Continuous case) If X has N(0,1), what is the moment
generating function of X ?
Properties
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For any constants a and b, the MGF of the random variable
aX + b is given by MaX +b (t) = ebt MX (at).
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For independent and identically distributed random
variables X , X1 , · · · , Xn . Let Sn = X1 + X2 + X3 + · · · + Xn .
Then MSn (t) = MXn (t), where MX (t) is the MGF of X .
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In particular, if X and Y are independent random variables
with MGFs MX (t) and MY (t) respectively, then the MGF of
X + Y is MX +Y (t) = MX (t)MY (t).
Examples
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(linear transformation) If X has N(µ, σ 2 ), what is the
moment generating function of X ?
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(Sum of two independent random variables) If X and Y are
independent Poisson random variables with parameters λ1
and λ2 respectively, what is the moment generating
function of X + Y ?
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(Sum of independent random variables) If X has
Binomial(n, p), what is the moment generating function of
X?