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Probability distributions 85
In contrast, in a continuous probability distribution the uncertain
quantity can take on any value within a specified interval. For example,
the time taken to assemble a component on a production line could take
on any value between, say, 0 and 30 minutes. There is no reason why
the time should be restricted to a whole number of minutes. Indeed, we
might wish to express it in thousandths or even millionths of a minute;
the only limitation would be the precision of our measuring instruments.
Because continuous uncertain quantities can, in theory, assume an
infinite number of values, we do not think in terms of the probability
of a particular value occurring. Instead, the probability that the variable
will take on a value within a given range is determined (e.g. what is the
probability that our market share in a year’s time will be between 5%
and 10%?). Figure 4.3 shows a probability distribution for the time to
complete a construction project. Note that the vertical axis of the graph
has been labeled probability density rather than probability because we
are not using the graph to display the probability that exact values will
occur. The curve shown is known as a probability density function (pdf).
The probability that the completion time will be between two values is
found by considering the area under the pdf between these two points.
Since the company is certain that the completion time will be between 10
and 22 weeks, the whole area under the curve is equal to 1. Because half
of the area under the curve falls between times of 14 and 18 weeks this
implies that there is a 0.5 probability that the completion time will be
between these two values. Similarly, 0.2 (or 20%) of the total area under
the curve falls between 10 and 14 weeks, implying that the probability
Probability density
Half of the total
area under the
curve Probability density
function (pdf)
10 14
18
Time to complete project (weeks)
Figure 4.3 – Probability distribution for project completion time
22