TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2022W/HW04.1

R-functions

(a)

trees is the R Dataset Package containing Diameter, Height and Volume for 31 Black Cherry trees. Using R define a vector $h$ that contains the values of the column Height and define a vector $v$ that contains the values of the column Volume from the dataset trees. Use the command table(h) to obtain the frequency table of the vector $h$. What are the outputs of the commands summary(h) and factor(h)? Compute the sample means mean() and the sample variances var() for both vectors. Use the command plot() to plot the points $(h,v)$.

(b)

Consider the curve given by the parametrization

${}t\mapsto \left({e^{-{\frac {t}{8}}}\cdot \cos {2t} \atop e^{\frac {t}{5}}\cdot \sin {3t}}\right){\text{for}}\ 0\leq t\leq 5\pi$

Use plot() to plot the curve and dev.print() to save the result as a pdf file.

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Lösung von Tutorin

(a)

trees
h=trees$Height
v=trees$Volume

factor(h)
# Output:
# [1] 70 65 63 72 81 83 66 75 80 75 79 76 76 69 75 74 85 86 71 64 78 80 74 72
# [25] 77 81 82 80 80 80 87
# Levels: 63 64 65 66 69 70 71 72 74 75 76 77 78 79 80 81 82 83 85 86 87

table(h)
# Output:
# 63 64 65 66 69 70 71 72 74 75 76 77 78 79 80 81 82 83 85 86 87
# 1  1  1  1  1  1  1  2  2  3  2  1  1  1  5  2  1  1  1  1  1

mean(h);mean(v)
# Output:
# [1] 76
# [1] 30.17097

var(h);var(v)
# Output:
# [1] 40.6
# [1] 270.2028

plot(h, v, type="p", xlab="Height (in ft)", ylab="Volume (in cubuc ft)", main="Volmes and Heights of 31 black cherry trees", col="blue")

(b)

plot.new()
par(mfrow=c(1,1))
t=seq(0,5*pi,0.01)
plot(exp(-t/8)*cos(2*t),exp(t/5)*sin(3*t), type='l', xlab="x", ylab="y", main = "Parametric curve", col="violet")

Lösungsvorschlag von Simplex

(a)

data(trees)
h <- trees [,2]
v <- trees [,3]
    
table_h <- table(h)
summary_h <- summary(h)
factor_h <- factor(h)
mean_h <- mean(h)
mean_v <- mean(v)
var_h <- var(h)
var_v <- var(v)
plot(h,v, type="l")

• summary(h): This command prints the following data retreived from the numbers: Mean, 1st Quadrand, Median, Mean, 3rd Quadrant and Max.
• factor(h): Prints the set (each number only one time) of all numbers sorted ascending.
• mean(): This function calculates the mean of the given list.
• var(): This function calculates the variance of the given list.

(b)

t <- seq(0,(5*pi),pi/50)
t_x <- exp(-t/8) * cos(2*t)
t_y <- exp(t/5) * sin(3*t)
plot(t_x, t_y, type="l")
dev.print(pdf, "print_1b")

--Simplex 22:43, 28. Jan. 2023 (CET)