4.3Jointly Distributed Random Variables 99
These probabilities are obtained as follows:
P{B=0,G= 0 }=P{no children}
=.15
P{B=0,G= 1 }=P{1 girl and total of 1 child}
=P{1 child}P{1 girl|1 child}
=(.20)
( 1
2
)
=.1
P{B=0,G= 2 }=P{2 girls and total of 2 children}
=P{2 children}P{2 girls|2 children}
=(.35)
( 1
2
) 2
=.0875
P{B=0,G= 3 }=P{3 girls and total of 3 children}
=P{3 children}P{3 girls|3 children}
=(.30)
( 1
2
) 3
=.0375
We leave it to the reader to verify the remainder of Table 4.2, which tells us, among other
things, that the family chosen will have at least 1 girl with probability .625. ■
We say thatXandYarejointly continuousif there exists a functionf(x,y) defined for
all realxandy, having the property that for every setCof pairs of real numbers (that is,
Cis a set in the two-dimensional plane)
P{(X,Y)∈C}=
∫∫
(x,y)∈C
f(x,y)dx dy (4.3.3)
The functionf(x,y) is called thejoint probability density functionofXandY.IfAandB
are any sets of real numbers, then by definingC={(x,y):x∈A,y∈B}, we see from
Equation 4.3.3 that
P{X∈A,Y∈B}=
∫
B
∫
A
f(x,y)dx dy (4.3.4)
Because
F(a,b)=P{X∈(−∞,a],Y∈(−∞,b]}
=
∫b
−∞
∫a
−∞
f(x,y)dx dy