TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW12.4

Correlation 4

Let $X\sim {\mathcal {N}}(2,1)$ and $Y=(X-2)^{2}$.

(a) Generate 200 realizations $(x_{i})_{i=1,\dots ,200}$ of the random variable $X$ and plot all $y_{i}=(x_{i}-2)^{2}$ against $x_{i},\ i=1,\dots ,200$.

(b) Compute the empirical correlation r of the realizetions $(x_{i})_{i}$ and $(y_{i})_{i}$.

(c) Would you say that a lack of correlation implies independence? What does the correlation measure? Would you say that the line is meaningful? Justify your answers.

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Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

# Let X ∼ N(2, 1) and Y = (X − 2)^2

# a) Generate 200 realizations (x_i)_{i=1,...,200} of the random variable X and plot all y_i = (x_i − 2)^2 against x_i, i = 1, . . . , 200.

rm(list = ls())

n <- 200
x <- rnorm(n, 2, 1)
y <- (x - 2)^2

plot(x, y, main="Corr of X and Y")

# b) Compute the empirical correlation r of the realizetions (x_i)_i and (y_i)_i

mean_x <- mean(x)
mean_y <- mean(y)
s_x <- sd(x)
s_y <- sd(y)

r <- (1 / (n - 1)) * sum((x - mean_x) * (y - mean_y)) / (s_x * s_y)

# c) Would you say that a lack of correlation implies independence? 
# What does the correlation measure? 
# Would you say that the line is meaningful? Justify your answers.

b_1 <- r * s_y / s_x
b_0 <- mean_y - b_1 * mean_x
abline(b_0, b_1)

# Not necessarily, it just measures how "strong" the linear relationship between the RVs is
# in this case the linear approximation is not usable but the variables may be quadratically correlated