Introduction to Probability and Statistics for Engineers and Scientists

4.3Jointly Distributed Random Variables 105

*4.3.2 Conditional Distributions

The relationship between two random variables can often be clarified by consideration of
the conditional distribution of one given the value of the other.
Recall that for any two eventsEandF, the conditional probability ofEgivenFis
defined, provided thatP(F)>0, by

P(E|F)=

P(EF)

P(F)

Hence, ifXandY are discrete random variables, it is natural to define the conditional
probability mass function ofXgiven thatY=y,by

pX|Y(x|y)=P{X=x|Y=y}

=

P{X=x,Y=y}

P{Y=y}

=

p(x,y)

pY(y)

for all values ofysuch thatpY(y)>0.

EXAMPLE 4.3f If we know, in Example 4.3b, that the family chosen has one girl, compute
the conditional probability mass function of the number of boys in the family.

SOLUTION We first note from Table 4.2 that

P{G= 1 }=.3875

Hence,

P{B= 0 |G= 1 }=

P{B=0,G= 1 }

P{G= 1 }

=

.10

.3875

=8/31

P{B= 1 |G= 1 }=

P{B=1,G= 1 }

P{G= 1 }

=

.175

.3875

=14/31

P{B= 2 |G= 1 }=

P{B=2,G= 1 }

P{G= 1 }

=

.1125

.3875

=9/31

P{B= 3 |G= 1 }=

P{B=3,G= 1 }

P{G= 1 }

= 0

Thus, for instance, given 1 girl, there are 23 chances out of 31 that there will also be
at least 1 boy. ■

* Optional section.