DATA 605 Assignment 5

Choose independently two numbers B and C at random from the interval [0, 1] with uniform density.

library(dplyr)

## Warning: package 'dplyr' was built under R version 3.5.3

library(ggplot2)

## Warning: package 'ggplot2' was built under R version 3.5.3

library(ggthemes)

## Warning: package 'ggthemes' was built under R version 3.5.3

set.seed(12345)
n <- 1000000
B <- runif(n, min = 0, max = 1)
C <- runif(n, min = 0, max = 1)
data <- data.frame(B = B, C = C)

1. Prove that B and C are proper probability distributions.

hist(data$B, main = "Histogram of B", xlab = "")

hist(data$C, main = "Histogram of C", xlab = "")

Note that the point (B,C) is then chosen at random in the unit square.

Find the probability that

2. B + C < 1/2 - Answer: The probability that B + C < 1/2 is 0.125

data <- data %>%
  mutate(a = ifelse(B + C < 1/2, 1, 0))
sum(data$a) / n

## [1] 0.125473

This is represented by the area in blue

ggplot(data, aes(x = B, y = C, color = as.factor(a))) +
  geom_point() +
  coord_cartesian(xlim = c(0, 1), ylim = c(0, 1)) +
  scale_color_fivethirtyeight() +
  theme_fivethirtyeight() +  ylab('C') + xlab('B') +
  theme(legend.position = "none", aspect.ratio=1) 

3. BC < 1/2 - Answers: The probability that BC < 1/2 is 0.85

data <- data %>%
  mutate(b = ifelse(B * C < 1/2, 1, 0))
sum(data$b) / n

## [1] 0.846341

ggplot(data, aes(x = B, y = C, color = as.factor(b))) +
  geom_point() +
  coord_cartesian(xlim = c(0, 1), ylim = c(0, 1)) + 
  theme(legend.position = "none", aspect.ratio=1) +
  scale_color_fivethirtyeight() +
  theme_fivethirtyeight() +  ylab('C') + xlab('B') +
  theme(legend.position = "none", aspect.ratio=1) 

4. |B - C| < 1/2 - Answer: The probability that |B - C| < 1/2 is 0.75

data <- data %>%
  mutate(c = ifelse(abs(B - C) < 1/2, 1, 0))
sum(data$c) / n

## [1] 0.749999

ggplot(data, aes(x = B, y = C, color = as.factor(c))) +
  geom_point() +
  coord_cartesian(xlim = c(0, 1), ylim = c(0, 1)) + 
  theme(legend.position = "none", aspect.ratio=1) +
  scale_color_fivethirtyeight() +
  theme_fivethirtyeight() +  ylab('C') + xlab('B') +
  theme(legend.position = "none", aspect.ratio=1) 

5. max(B, C) < 1/2 - Answer: The probability that max(B, C) < 1/2 is 0.25

data <- data %>%
  mutate(d = ifelse(pmax(B, C) < 1/2, 1, 0))
sum(data$d) / n

## [1] 0.250294

ggplot(data, aes(x = B, y = C, color = as.factor(d))) +
  geom_point() +
  coord_cartesian(xlim = c(0, 1), ylim = c(0, 1)) + 
  theme(legend.position = "none", aspect.ratio=1) +
  scale_color_fivethirtyeight() +
  theme_fivethirtyeight() +  ylab('C') + xlab('B') +
  theme(legend.position = "none", aspect.ratio=1) 

6. min(B, C) < 1/2 - Answer: The probability that min(B, C) < 1/2 is 0.75

data <- data %>%
  mutate(e = ifelse(pmin(B, C) < 1/2, 1, 0))
sum(data$e) / n

## [1] 0.749954

ggplot(data, aes(x = B, y = C, color = as.factor(e))) +
  geom_point() +
  coord_cartesian(xlim = c(0, 1), ylim = c(0, 1)) + 
  theme(legend.position = "none", aspect.ratio=1) +
  scale_color_fivethirtyeight() +
  theme_fivethirtyeight() +  ylab('C') + xlab('B')