DATA 605 Week 7 - Chapter 5.2 - Question 12 (p.220)

Chapter 5.2 - Question 12

A number \(U\) is chosen at random in the interval [0,1]. Find the probability that:

1. \(R = U^2 < 1/4\)
2. \(S = U(1-U) < 1/4\)
3. \(T = U/(1-U) < 1/4\)

First we’ll generate 1M observations for \(U\). We’ll also visually check to ensure that the distrubution is uniform:

n <- 1000000
U <- (matrix(sample(c(0:n),size = n, replace = T), nrow=n, ncol=1))/n
hist(U)

The distribution looks okay, so now we can find the probabilities of \(R,S\) & \(T\)

#for R in part A
R <- sum(U^2 < 0.25)/n

#for S in part B
S  <- sum(U * (1-U) < 0.25) / n

#for T in part C
T <- sum(U /(1-U) < 0.25) / n