Page 95 - Fundamentals of Probability and Statistics for Engineers

78 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
52 52
Z Z rf r†
EfRjAgˆ rf rjA†dr ˆ R dr:
R
48 48 c
This integral can be evaluated when f (r) is specified.
R
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
1
P X x 0 †ˆ :
2
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
speak of median income as a good central measure of personal 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.

Example 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,
t
e ; for t 0;
f t†ˆ
T
0; elsewhere;
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

Z 1
f t†dt ˆ ;
T
0 2
or

2

ˆ ln :

TLFeBOOK