M348
Homework #7
Burden and Faires.
Section 4.3 (#16, 20). Section 4.4 (#2a, 4a, 6a, 12a). Section 4.5 (#2a).
Ortega and Grimshaw.
Chapter 14: #14.3, 14.6, 14.13∗.
*Omit the declaration const in the template definition.
Programming mini-project.
Suppose the heights of individuals in a population are described by a Gaussian distribution with mean
µand standard deviation σ > 0. Then the probability that the height xof a randomly chosen individual
lies in an interval (a,b) is given by
P{a < x < b}=1
σ√2πZb
a
e−(x−µ)2/2σ2dx.
Here we use a composite integration rule to approximate this probability for different intervals.
(a) Use the change of variable u= (x−µ)/σ in the above integral to show that
P{a < x < b}=1
√2πZ(b−µ)/σ
(a−µ)/σ
e−u2/2du.
Moreover, show that P{µ−cσ < x < µ +cσ}is independent of µand σand depends only on c.
(b) Write a C++ program to approximate RB
Af(u)du using the Composite Simpson Rule, where f(u)
may be any user-defined function. For a given f(u), your program should accept the interval
parameters Aand B, an initial number of subintervals n(even) and a tolerance parameter tol . The
program should compute approximations with larger and larger nuntil the difference between two
consecutive approximations is less than t ol in absolute value.
(c) Use your program to compute the probabilities P{µ−σ < x < µ +σ},P{µ−2σ < x < µ + 2σ}
and P{µ−3σ < x < µ + 3σ}, each to within 10−5.