Engineering Economic Analysis

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Joint Probability Distributions

CombiningJhese, we write

2P(9) +P(9) __ t

P(9) 1/3

P(6) =2/3

The probability distribution for thelifejs P(6) ... 6,6.7%andP(9)- 33.3%.

JOINT PROBABILITY DISTRIBUTIONS

Example10-5 constructed the probability distributions for a project's annual benefit and

life. These examples show how likely each value is for the input data of the problem. .We

would like to construct a similar probability distribution for the project's present worth. This

is the distribution that we can use to evaluate the project. That present worth depends on

both input probability distributions, so we need to construct thejoint probability distribution

for the different combinations of their values.

For this introductory text,we assume that two random variables such as the an-

nual benefit and life are unrelated or statistically independent. This means that thejoint

probability of a combined event (Event A defined on the first variable and Event B on

the second variable) is the product of the probabilities for the two events. This is

Equation 10-4:

If A and B are independent, thenpeAand B)=peA) xPCB) (10-4)

For example, flipping a coin and rolling a die are statistically independent. Thus, the prob-

abilityof {flippinga head and rollinga 4} equalsthe probabilityof a {heads}=Y2times

the probability of a {4}=1/6, for a joint probability =1/12.

. The number of outconies in the joint distribution is the product of the number of
outcomes in each variable's distribution. Thus, for the coin and the die, there are 2 times
6 or 12 combinations. Each of the 2 outcomes for the coin is combined with each of the 6
outcomes for the die.
Some variables are not statistically independent, and the calculation of their joint prob-
ability distribution is more complex. For example, a project with low revenues may be
terminated early and one with high revenues may be kept operating as long as possible.
In these cases annual cash flow and project life are not independent. While this type of
relationshipcan sometimesbemodeledwith economicdecisiontrees (coveredlater:in
this chapter),we will limit our coveragein this text to the simplercase of independent
variables.
Example 10-6uses the three values and probabilities for the annual benefit and the two
values and probabilities for the life to construct the six possible combinations. Then the
values and probabilities are constructed for the project's PW.

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