TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2022W/HW05.5

Histograms of averages of exp(1)

(a) Generate a frequency histogram of 1000 samples from an exp(1) random variables.

(b) Generate a density histogram for the average of 2 independent exp(1) random variable.

(c) Using rexp(), matrix() and colMeans() generate a density histogram for the average of 50 independent exp(1) random variables. Make 10000 sample averages and use a binwidth of 0.1 for this. Look at the spread of the histogram.

(d) Add a graph of the pdf of ${\mathcal {N}}(1,1/50)$ on your plot in problem (c).

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Lösungsvorschlag von Friday

--Friday Sa 30 Jan 2021 17:03:56 CET

# Statistics and Probability HW #5
# Friday 
# Duedate: 09.11.2020

# Problem 6 - Histograms of averages of exp(1)

# Problem 6a)
# Generate a frequency histogram of 1000 samples from an exp(1) random 
# variables.
d = rexp(1000,1)
hist(d, freq=TRUE, xlab="", ylab="frequency", main="Frequency Histogram")

# Problem 6b)
# Generate a density histogram for the average of 2 independent exp(1) random 
# variable.
d1 = rexp(1000,1)
d2 = rexp(1000,1)
d = (d1 + d2) / 2
hist(d, freq = TRUE, xlab="", ylab="density", main="Desity Histogram")

# Problem 6c)
# Using rexp(), matrix() and colMeans() generate a density histogram for the 
# average of 50 independent exp(1) random variables. Make 10000 sample averages 
# and use a binwidth of 0.1 for this. Look at the spread of the histogram.
exp = 50
n = 10000
x = rexp(n*exp,1)
tmp = matrix(x, nrow=exp, ncol=n)
d = colMeans(tmp)
hist(d, freq=FALSE, xlab="", ylab="density", main="Desity Histogram")

# Problem 6d)
# Add a graph of the pdf of N(1, 1/50) on your plot in problem (c).
d = seq(0,2,.01)
lines(d,dnorm(d,1,1/sqrt(50)),lwd=3)

Lösungsvoschlag von Simplex

# a)
samples = rexp(1000, 1)
hist(samples)

# b)
var1 <- rexp(1000, 1)
var2 <- rexp(1000, 1)
avg <- (var1 + var2)/2
hist(avg, freq=FALSE)

# c)
mat <- matrix(nrow=50, ncol=10000)
for (i in 1:50)
  mat[i,] <- rexp(10000, 1)
avg <- colMeans(mat)

bin_width <- 0.1
bins <- seq(min(avg), max(avg) + bin_width, bin_width)
hist(avg, breaks=bins, freq=FALSE)

# d)
sd <- 1/50
x <- seq(1 - 3*sqrt(sd), 1 + 3*sqrt(sd), 0.001)
norm_values <- dnorm(x, 1, sqrt(sd))
lines(x, norm_values)

--Simplex 15:25, 3. Feb. 2023 (CET)