Fundamentals of Probability and Statistics for Engineers

Answer: the average value required in this example is a conditional mean of R
given the event A. Although no formal definition is given, it should be clear that
the desired average is obtained from

This integral can be evaluated when fR(r) is specified.

Two other quantities in common usage that also give a measure of centrality
of a random variable are its median and mode.
A median of X is any point that divides the mass of the distribution into two
equal parts; that is, x 0 is a median of X if

The mean of X may not exist, but there exists at least one median.
In comparison with the mean, the median is sometimes preferred as a
measure of central tendency when a distribution is skewed, particularly where
there are a small number of extreme values in the distribution. For example, we
sp eak of median income as a good central measure of per sonal income for a
population. This is a better average because the median is not as sensitive to
a small number of extremely high incomes or extremely low incomes as is
the mean.

Ex ample 4. 4. Let T be the time between emissions of particles by a radio-
active atom. It is well established that T is a random variable and that it obeys
an exponential distribution; that is,

where is a positive constant. The random variable T is called the lifetime of
the atom, and a common average measure of this lifetime is called the half-life,
which is defined as the median of T. Thus, the half-life, is found from

or

78 Fundamentals of Probability and Statistics for Engineers

EfRjAgˆ

Z 52

48

rfR…rjA†drˆ

Z 52

48

rfR…r†

c

dr:

P…Xx 0 †ˆ

1

2

:

fT…t†ˆ

et; fort 0 ;

0 ; elsewhere;



Z

0

fT…t†dtˆ

1

2

;

ˆln

2





: