Homework Set 4, Due Thurs. Oct. 9 in Class
(Late papers OK until 1:00 Friday under Ms. Li's door, DRL 2C11)
- a). Choose a number c at random from the interval [0,1]. Find the
probability that:
i). |c - 0.5| < 0.25 ii). 3c2 < c
b). Simultaneously pick two numbers (r,s) at random from the interval [0,1]. Compute the probability that:
i) r + s < 1/2 ii). |r - s| < 1/2 iii). Both r < 1/2 and 1 - s < 1/2
iv). |r - s| < 1/2 and r < 1/2 and 1 - s < 1/2
v). r2 + s2 < 1/2
Suggestion: (r,s) may be viewed as a point in the unit square. - Say 48 different people measure the height of a building,
with the resulting measurements p1,
p2,. . .,p48. One standard way to compute the
"average" P is to pick P so that it minimizes
the error:
Q(P) = (p1 - P)2 + . . . + (p48 - P)2 a). Show that P = (p1 + . . . + p48)/48
b). If you change one measurement, say replacing p2 by p2 + c, what is the new value of P? - A complication. In the previous problem, we believe that all of
the observers are equally reliable, except for one of them, say the
person reporting p48. We thus want that less reliable
measurement to have less influence on the "average". We use a
weighted average using a different weight only for the less trusted
observer:
S(P) = a(p1 - P)2 + a(p2 - P)2 + . . . + a(p47 - P)2 + b(p48 - P)2, where the weights a > 0 and b > 0 are to be found.
a). Compute the average P to minimize S(P). [Your answer will involve a and b.]
b). If you change only one of the more trusted measurements, say replacing p2 by p2 + c, what is the new value of P?
c). If you change only the less trusted measurement, p48, say replacing p48 by p48 + c, what is the new value of P?
d). Say you'd like the average P to be half as sensitive to the data p48. Thus you want the change in the value of part c) to be half that of part b). Find the weights a and b. - a). When I taught first year calculus recently, I gave three hour
exams plus homework and a final exam. Here are the
raw hour exam
scores. To preserve privacy, I used names like "S27" for the
students. This table contains a summary:
Exam 1 Exam 2 Exam 3 mean 75.3 55.9 67.6 std deviation 14.6 12.8 18.5 Exam 2 was much more difficult. Histograms add insight:
Exam 1 Exam 2 Exam 3 The grades of student "S40" (look at the data) are: 85 76 77 so her lowest exam score was on the 2nd hour exam. She noted that that the 2nd hour exam was the most difficult yet her score was one of the highest grades in the class. It seems both incorrect and unfair not to take that into consideration in computing her course average.
Can you devise some better way to compute course averages?b). In another course, to compute the final course grade, I told the class I would drop the lowest score from their hour exams. Thus, I would be counting only the two best hour exams. Say we do this with this class' exam grades.
Using the same data as above, student "S40" might have noted that since the 2nd hour exam was the most difficult, and her grade was one of the highest in the class, it seems both incorrect and unfair to drop her score of 76. Can anything useful be done to resolve this?Method of Least Squares
(Reference: Vectors and Least Squares) - Use the Method of Least Squares to find the straight line that best
fits the following data given by the points (x,y):
x -2 -1 0 2 y 4 3 1 0 - The water level in the North Sea is mainly determined by the
so-called M2 tide, whose period is about 12 hours. The height H(t)
thus roughly has the form
where time t is measured in hours. Say one has the following measurements:
t (hours) 0 2 4 6 8 10 H(t) (meters) 1.0 1.6 1.4 0.6 0.2 0.8 Use the method of least squares with these measurements to find the constants a, b, and c in H(t) for this data.
Note: the method of least squares cannot be applied to the mathematicall equivalent expression:
A newspaper article you may find interesting: Chatterbots: The Loebner Prize