Homework 1 ECE 4710J

Homework 1 ECE 4710J

1. Fundamental Linear Algebra
Ben, Tom, and Amy are shopping for fruit at a grocery store. A fruit bowl contains some fruit
and the price of fruit bowl is the total price of all of its individual fruit.
The store has apples for $2, bananas for $1, and oranges for $4. The price of each of these can
be written in a vector:
−→v =


2
1
4


The store sells the following fruit bowls:
1. 2 of each fruit
2. 5 apples and 8 bananas
3. 2 bananas and 3 oranges
4. 10 oranges
(a) Define a matrix B such that
B
−→v
evaluates to a length 4 column vector containing the price of each fruit bowl. The first
entry of the result should be the cost of fruit bowl 1, the second entry the cost of fruit
bowl 2, etc.
(b) Ben, Tom, and Amy make the following purchases:
• Ben buys 2 fruit bowl 1s and 1 fruit bowl 2.
• Tom buys 1 of each fruit bowl.
• Amy buys 10 fruit bowl 4s
Define a matrix A such that the matrix expression
AB−→v
evaluates to a length 3 column vector containing how much each of them spent. The first
entry of the result should be the total amount spent by Ben, the second entry the amount
sent by Tom, etc.
(c) Let’s suppose the store changes their fruit prices, but you don’t know what they changed
their prices to. Ben, Tom, and Amy buy the same quantity of fruit baskets and the number
of fruit in each basket is the same, but now they each spent these amounts:
−→x =


80
80
100


In terms of A, B, and −→x , determine −→v2 (the new prices of each fruit).
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2. Calculus
Let σ(x) = 1
1+e−x .
(a) Show that σ(−x) = 1 − σ(x)
(b) show that the derivative can be written as:
d
dx σ(x) = σ(x)(1 − σ(x))
(c) Make a plot for σ(x) and d
dx σ(x) in the same coordinate system for x ∈ [−5, 5]
3. Minimization
Consider the function f(c) = 1
nΣ
n
i=1(xi − c)
2
. In this scenario, suppose that our data points
x1, x2, ..., xn are fixed, and that c is the only variable.
Using calculus, determine the value of c that minimizes f(c). You must justify that this is
indeed a minimum, and not a maximum.
4. Probability
Only 1% of 40-year-old women who participate in a routine mammography test have breast
cancer. 80% of women who have breast cancer will test positive, but 9.6% of women who don’t
have breast cancer will also get positive tests.
Suppose we know that a woman of this age tested positive in a routine screening. What is
the probability that she actually has breast cancer? (Note: You must show all of your work,
and also simplify your final answer to 3 decimal places.)
5. Statistics
Suppose we collected a sample of 200 students at University A, and 150 of them happened to
be Canadian (so, if we were to select a student uniformly at random from our sample, there is
a 0.75 chance that they are Canadian).
For inferential purposes, we choose to bootstrap this sample 500,000 times. That is, we simulate
the act of re-sampling (with replacement) 200 students from our observed sample, and each time
we record the number of Canadians in our re-sample. We provide a histogram of the sampling
distribution below.
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What is the standard deviation of the sampling distribution shown above? Select the closest
option below, and explain your answer.
a 1.5
b 6.1
c 12.4
d 10.1
Hint: While it is possible to calculate the answer, the histogram has all of the information you
need.
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