Introduction to Probability and Statistics for Engineers and Scientists

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