QBUS1040 Assignment 1 Semester 1, 2022
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QBUS1040 Assignment 1
Semester 1, 2022
1. (60 points) DataCamp Python exercises. We have set up a dedicated QBUS1040 group on DataCamp. You can join with the link we provided on Canvas. Please make sure you use your university email address (@uni.sydney.edu.au) for your registration. Otherwise, we will not be able to identify and count your work towards your grades.
2. Vectors of second diﬀerences. Suppose x is an n-vector. The associated vector of diﬀerences is the (n - 1)- vector d given by d = (x2 - x1 , x3 - x2 , . . . , xn - xn − 1 ).
(a) (5 points) Express d in terms of x using vector operations (e.g., slicing notation, sum, diﬀerence, linear
combinations, inner product). The diﬀerence vector has a simple interpretation when x represents a time series. For example, if x gives the daily value of some quantity, d gives the day-to-day changes in value of the quantity.
(b) (5 points) The vector of second diﬀerence is (n - 2)-vector s given by s = (d2 - d1 , d3 - d2 , . . . , dn − 1 -
3. Average weight in a population. Suppose the 200-vector x represents the distribution of weight in some population of people, where xi is the number of people who weigh i kilograms. (You can assume the weight is measured as a discrete number, x 0, and that no one in the population weighs over 200 kilograms.) Find expressions, using vector notation, for the following quantities.
(a) (4 points) The total number of people in the population.
(b) (4 points) The total number of people in the population whose weight is 70 kg and above.
4. (10 points) Taylor approximation. Consider the function f : R3 → R given by f (x1 , x2 , x3 ) = 3x1(2)x2 + 4x2(2)x3 . Find the Taylor approximation of fˆ at the point z = (1, 1, -1). Compare f (x) and fˆ(x) for the following values of x:
x = (1, 1, -1), x = (2, 1, 0), x = (-1, 2, 1), x = (4, -4, 4)
(a) (4 points) Either give the value of f (1, -0.5, -1), or state why it cannot be determined. (b) (4 points) Either give the value of f (2, -2, 0), or state why it cannot be determined.