Chapter 2
Multivariate Distributions
2.1 DistributionsofTwoRandomVariables
We begin the discussion of a pair of random variables with the following example. A
coin is tossed three times and our interest is in the ordered number pair (number of
H’s on first two tosses, number of H’s on all three tosses), where H and T represent,
respectively, heads and tails. LetC={TTT,TTH,THT,HTT,THH,HTH,HHT,
HHH}denote the sample space. LetX 1 denote the number of H’s on the first
two tosses andX 2 denote the number of H’s on all three flips. Then our inter-
est can be represented by the pair of random variables (X 1 ,X 2 ). For example,
(X 1 (HTH),X 2 (HTH)) represents the outcome (1,2). Continuing in this way,X 1
andX 2 are real-valued functions defined on the sample spaceC, which take us from
the sample space to the space of ordered number pairs.
D={(0,0),(0,1),(1,1),(1,2),(2,2),(2,3)}.ThusX 1 andX 2 are two random variables defined on the spaceC, and, in this
example, the space of these random variables is the two-dimensional setD,whichis
a subset of two-dimensional Euclidean spaceR^2. Hence (X 1 ,X 2 ) is a vector function
fromCtoD. We now formulate the definition of a random vector.
Definition 2.1.1(Random Vector). Given a random experiment with a sample
spaceC, consider two random variablesX 1 andX 2 , which assign to each element
cofCone and only one ordered pair of numbersX 1 (c)=x 1 ,X 2 (c)=x 2 .Then
we say that(X 1 ,X 2 )is arandom vector.Thespaceof(X 1 ,X 2 )is the set of
ordered pairsD={(x 1 ,x 2 ):x 1 =X 1 (c),x 2 =X 2 (c),c∈C}.
We often denote random vectors using vector notationX=(X 1 ,X 2 )′,where
the′denotes the transpose of the row vector (X 1 ,X 2 ). Also,weoftenuse(X, Y)
to denote random vectors.
LetDbe the space associated with the random vector (X 1 ,X 2 ). LetAbe a
subset ofD. As in the case of one random variable, we speak of the eventA.We
wish to define the probability of the eventA, which we denote byPX 1 ,X 2 [A]. As
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