SRout_Assignment_5

Choose independently two numbers B and C at random from the interval [0, 1] with uniform density. Prove that B and C are proper probability distributions. Note that the point \((B,C)\) is then chosen at random in the unit square.

Solution:

Create two random variables i.e B and C, by sampling 10,000 numbers between 0 and 1.

library(ggplot2)
set.seed(123)
n <- 10000
B <- runif(n, min = 0, max = 1)
C <- runif(n, min = 0, max = 1)
hist(B)

hist(C)

Histogram of B and C shows uniform density diribution between 0 and 1.

Find the probability that \((a)\) B + C < 1/2

Solution:

sum_BC <- B + C
pr_B_plus_C <- sum(sum_BC < 0.5) / n
pr_B_plus_C

## [1] 0.1217

# density plot of B + C
ggplot(as.data.frame(sum_BC), aes(sum_BC)) + 
  geom_density(colour="red") +
  labs(title = "Density Plot of B + C", x = "Sum of Random Values") +
  theme(plot.title = element_text(hjust = 0.5))

Find the probability that \((b)\) BC < 1/2

Solution:

mul_BC <- B * C
pr_B_mul_C <- sum(mul_BC < 0.5) / n
pr_B_mul_C

## [1] 0.8562

# density plot of B and C
ggplot(as.data.frame(mul_BC), aes(mul_BC)) + 
  geom_density(colour="red") +
  labs(title = "Density Plot of B and C", x = "Product of Random Values") +
  theme(plot.title = element_text(hjust = 0.5))

Find the probability that \((c)\) |B − C| < 1/2

Solution:

abs_BC <- abs(B - C)
pr_B_abs_C <- sum(abs_BC < 0.5) /n
pr_B_abs_C

## [1] 0.747

# density plot of abs(B and C)
ggplot(as.data.frame(abs_BC), aes(abs_BC)) + 
  geom_density(colour="red") +
  labs(title = "Density Plot of |B - C|", x = "Difference of Random Values") +
  theme(plot.title = element_text(hjust = 0.5))

Find the probability that: \((d)\) max{B,C} < 1/2

Solution:

max_BC <- pmax(B,C)
pr_B_max_C <- sum(max_BC < 0.5) /n
pr_B_max_C

## [1] 0.2558

# density plot of max(B and C)
ggplot(as.data.frame(max_BC), aes(max_BC)) + 
  geom_density(colour="red") +
  labs(title = "Density Plot of Max(B and C)", x = "Maximum of Random Values") +
  theme(plot.title = element_text(hjust = 0.5))

Find the probability that: \((e)\) min{B,C} < 1/2

Solution:

min_BC <- pmin(B,C)
pr_B_min_C <- sum(min_BC < 0.5) /n
pr_B_min_C

## [1] 0.7597

# density plot of min(B and C)
ggplot(as.data.frame(min_BC), aes(min_BC)) + 
  geom_density(colour="red") +
  labs(title = "Density Plot of Min(B and C)", x = "Minimum of Random Values") +
  theme(plot.title = element_text(hjust = 0.5))