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where f isgivenby(1.1). And this is exactly the problem we want to solve in this
chapter: How do we obtain a numerical value for p, given a value of the number
of bagels b? The reason for us to seek a numerical solution is that (1.2) cannot be
integrated analytically when f is of the form (1.1).
1.2
The Computational Problem
As stated above, the problem we want to solve is to compute the probability p given
by (1.2) for given values of the number of bagels b. Before we introduce a numerical
method for this problem, we can do some observations that will simplify our task.
Our problem is to estimate the number of bagels b such that the owner can meet
the demand on at least 95% of the days. Thus we want to find b such that
p
>
0:95:
(1.3)
In order to do this, we have to be able to compute p givenby(1.2), for any rele-
vant value of the upper limit b. We will devise a numerical algorithm for computing
such an approximation. One main step in deriving this algorithm is to divide the
interval into a number of small subintervals and then approximate the integral on
each of these smaller domains. However, negative infinity
1
as the lower limit is
difficult to deal with, because it will result in an infinite number of subintervals. We
therefore want to rephrase the problem in terms of a definite integral on a bounded
domain. In order to do this, we start by recalling that the average number of bagels
sold each day is 300. Thus, if we have chosen b
D
300, we would only have a
sufficient supply on 50% of the days. Consequently, we only consider the case of
b > 300:
(1.4)
Now, we can divide the integral of (1.2) into two parts:
p
D
Z
300
1
f.x/dx
C
Z
b
300
f.x/dx:
(1.5)
Here we can compute the first term analytically. To do this, we first note that f is
symmetric, in the sense that
f .300
C
x/
D
f .300
x/
for all x. It follows from this property that
Z
300
f.x/dx
D
Z
1
300
f.x/dx;
1
