Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.1. Distributions of Two Random Variables 89

x y

z

Figure 2.1.1: A sketch of the the surface of the joint pdf discussed in Example

2.1.3. On the figure, the origin is located at the intersection of thexandzaxes

and the grid squares are 0.1 by 0.1, so points are easily located. As discussed in the

text, the peak of the pdf occurs at the point (

√

2 / 2 ,

√

2 /2).

Likewise we may extend the pmfpX 1 ,X 2 (x 1 ,x 2 ) over a convenient set by using zero
elsewhere. Hence, we replace
∑∑

D

pX 1 ,X 2 (x 1 ,x 2 )by

∑

x 2

∑

x 1

p(x 1 ,x 2 ).

2.1.1 MarginalDistributions......................

Let (X 1 ,X 2 ) be a random vector. Then bothX 1 andX 2 are random variables.
We can obtain their distributions in terms of the joint distribution of (X 1 ,X 2 )as
follows. Recall that the event which defined the cdf ofX 1 atx 1 is{X 1 ≤x 1 }.
However,

{X 1 ≤x 1 }={X 1 ≤x 1 }∩{−∞<X 2 <∞}={X 1 ≤x 1 ,−∞<X 2 <∞}.

Taking probabilities, we have

FX 1 (x 1 )=P[X 1 ≤x 1 ,−∞<X 2 <∞], (2.1.7)