Description
1. Use integration by parts to evaluate the following indefinite integral.
Z
x e2x
dx
A general rule for integration by parts is: R
f(x)g
0
(x)dx = f(x)g(x) −
R
f
0
(x)g(x)dx.
Letting f(x) = x and g
0
(x) = e
2x
in this rule gives
Z
x e2x
dx =
x e2x
2
−
Z
e
2x
2
dx
=
x e2x
2
−
e
2x
4
+ c (c ∈ R).
2. Let w = {9, 27, 15, 8, 10, 2, 70, 1, 4, 17, 9, 44}.
a) Compute the five number summary of w. Do this two ways: (1) by hand using the
definition on slide 44 and (2) using R. Make sure you get the same answers.
(1) To find the five number summary by hand, first we rewrite w in ascending order.
w = {1, 2, 4, 8, 9, 9, 10, 15, 17, 27, 44, 70}.
From this we see the minimum is 1 and the maximum is 70. Since w has an even
number of points, when finding the median we select the two values in the middle of
the sorted set which are 9 and 10. We take the smaller of the two so that half of the
values are less than or equal to 9- so hatq0.5 = 9. We then apply this same process to
the lower values of w which are {1, 2, 4, 5, 9}. We find that 4 and 5 are in the middle
and take the smaller of the two, 4, as the 25th percentile point. Similarly with the
values greater than the median, {10, 15, 17, 27, 44, 70} we find that the 75th percentile
data point is 17. This yields the five number summary:
min(w) = 1, qˆ0.25 = 4, qˆ0.5 = 9, qˆ0.75 = 17, max(w) = 70.
(2) To find the five number summary using R, the following command works and
produces the output shown:
quantile(w, type = 1)
0% 25% 50% 75% 100%
1 4 9 17 70
b) Draw (with a pen/pencil) a box plot summarizing w and comment on the shape of
the distribution (is it symmetric, are there any atypical observations, etc.). Be sure to
state what numerical values you use for the limits of the box, the limits of the whiskers,
and the median (and how they were calculated).
The limits of the box are the 1st and 3rd quartile markers, 4 and 17. The bar in the
middle of the box is ˆq0.5. The whiskers limits were calculated to be -15.5 and 36.5. The
smallest and largest values in the interval [−15.5, 36.5] were 1 and 27. These two numbers
are where the whisker bars are made. There are two values outlying these bounds, 44 and
70, which are plotted individually. The distribution is not symmetric. We observe that
the data is generally skewed to the right (or upward in the way we created the box plot)
by the outliers 70 and 44. We note that the mean of the data is 18, which is nearly double
qˆ0.5 = 9. The values less than ˆq0.5 are much more closely grouped than the values greater
than ˆq0.5.
3. List the changes to get from slide 65 to 66 in the slide deck EDA1.pdf and explain how
each change improves the visual comparison of the two density estimates.
The Density axes are changed so that the vertical axis of each figure matches the height
and formatting style of the other, since the Density markers are at different heights with
respect to the data on the first slide. An extra tick is also added so that the data does
not exceed the highest density marker. Also the histogram formatting is changed to a
box so that both figures enclose their data in a box of the same size- which makes it
easier to sense the similarity of the two types of plots. This also eliminates the title from
the histogram, which was not needed since it is clearly a histogram and also so that it
is not the case that one figure has a title and the other does not. There is some spacing
added below the 0 point of the density axis of the histogram, which helps it match the
kernel density figure. Similarly to the vertical axis, the horizontal axis of the histogram
is changed to match the style of the kernel density estimation figure. This again helps
to avoid differences in the formatting so that the focus is on the differences in plotting
results. Finally the horizontal axis label is changed to ”2012 Citigroup Closing Price” in
both figures so that they match and to avoid unnecessary information being included in
the figures.
4. Choose a ticker symbol: if your initials are a ticker symbol, use that; otherwise choose
a publicly traded company that starts with the same letter as your first name and use
its ticker symbol. Download closing price data for 2012 and recreate the visual display
in slide 65. You may need to adjust the bandwidth of the kernel density estimate if the
automatic choice is not good. The answer should include both the visual display and the
R code used to create it.
The R code will be provided in separate file so that you may run it on your own machine
and see the results. I include the plots here as well, which do match slide 65 exactly except
for the fact that citi was not used in my case.