Statistics 100B
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University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou Joint moment generating functions
t1
t2
Let X = . , be a random vector and let t = . . The joint moment generating
X2
. .
Xn tn function of X is defined as MX(t) = Eet′X = Eexp(ni=1 tixi).
Theorem
Let M (t) = ∂MX(t), M (t) = ∂2MX(t), and M (t) = ∂2MX(t).
i ∂ti ii ∂ti2 ij ∂ti∂tj Then, EXi = Mi(0), EXi2 = Mii(0), and EXiXj = Mij(0).
Corollary
Let ψ(t) = logM (t), ψ (t) = ∂ψX(t), ψ (t) = ∂2ψX(t), and ψ (t) = ∂2ψX(t).
X i ∂ti ii ∂ti2 ij ∂ti∂tj Then EXi = ψi(0), var(Xi) = ψii(0), and cov(XiXj) = ψij(0).
Theorem Y
Let X = Z . The marginal moment generating function of Y (Z) is the moment gen- erating function of X ignoring the vector Z (Y). This is expressed as MY(u) = MX(u,0)
u and MZ(v) = MX(0,v), where t = v .
Proof
Theorem
If Y and Z are independent then MX(t) =MY(u)MZ(v). Proof
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Example 1
X1
X = X2 have joint moment generating function
X3
MX(t1, t2, t3) = (1 − t1 + 2t2)−4(1 − t1 + 3t3)−3(1 − t1)−2.
Use the corollary on page 1 to find:
a. E(X1), E(X2), E(X3).
b. var(X1), var(X2), var(X3).
c. cov(X1, X2), cov(X1, X3), cov(X2, X3). d. ρX1,X3.
Example 2
Let X and Y be independent normal random variables, each with mean μ and standard deviation σ.
a. Consider the random quantities X + Y and X − Y . Find the moment generating function of X + Y and the moment generating function of X − Y .
b. Find the joint moment generating function of (X + Y, X − Y ).
c. Are X + Y and X − Y independent? Explain your answer using moment generating
functions.
Example 3
Let X = (X1, X2, X3) has joint moment generating function
MX(t1, t2, t3) = (1 − t1 + 2t2)−4(1 − t1 + 3t3)−3(1 − t1)−2. Answer the following questions:
a. Find the moment generating function of (X1,X3). b. Find the moment generating function of X1.
c. Find the moment generating function of X3.
d. Are X1, X3 independent?
e. Find the moment generating function of (X2,X3).
f. Are X2, X3 independent?
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