Fundamentals of Probability and Statistics for Engineers

Example 3.6.Let us discuss again Example 2.11 in the context of random
variables. Let X be the random variable representing precipitation levels, with
values 1, 2, and 3 indicating low, medium, and high, respectively. The random
variable Y will be used for the peak flow rate, with the value 1 when it is cr itical
and 2 when noncritical. The information given in Example 2.11 defines jpmf
pX Y (x,y), the values of which are tabulated in Table 3.1.
In order to determine the probability of reaching the critical level of peak
flow rate, for example, we simply sum over all pX Y (x, y) satisfying y 1,
regardless of x values. Hence, we have

The definition of jpmf for more than two random variables is a direct extension
of that for the two-random-variable case. Consider n random variables
X 1 ,X 2 ,...,Xn. Their jpmf is defined by

which is the probability of the intersection of n events. Its properties and
utilities follow directly from our discussion in the two-random-variable case.
Again, a more co mpact form for the jp mf is pX (x)whereX is an n-dimensional
random vector with co mponents X 1 ,X 2 ,...,Xn.

3.3.3 Joint Probability D ensity F unction

As in the case of single random variables, probability density functions become
appropriate when the random variables are co ntinuous. The joint probability

Table 3.1 Joint probability mass function for low, medium, and high precipitation
levels (x 1, 2, and 3, respectively) and critical and noncritical peak flow rates (y 1
and 2, respectively), for Example 3.6

yx

12 3

1 0.0 0.06 0.12
2 0.5 0.24 0.08

Random Variables and Probability D istributions 55

ˆ ˆ

P…X>Y†ˆP…Xˆ 5 \Yˆ 0 †‡P…Xˆ 4 \Yˆ 1 †‡P…Xˆ 3 \Yˆ 2 †

ˆ 0 : 01024 ‡ 0 : 0768 ‡ 0 : 2304 ˆ 0 : 31744 :

ˆ

P…Yˆ 1 †ˆpXY… 1 ; 1 †‡pXY… 2 ; 1 †‡pXY… 3 ; 1 †ˆ 0 : 0 ‡ 0 : 06 ‡ 0 : 12 ˆ 0 : 18 :

pX 1 X 2 ...Xn…x 1 ;x 2 ;...;xn†ˆP…X 1 ˆx 1 \X 2 ˆx 2 \...\Xnˆxn†; … 3 : 23 †