Data 605 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. Find the probability that

(a) B + C < 1/2

B=seq(0,1,length=2000)
C=seq(0,1,length=2000)
y=dnorm(punif(B, min = 0, max = 1) + runif(C, min = 0, max = 1))
boxplot(subset(y,y<.5),type="l", lwd=2, col="blue")

length(subset(y,y<.5))/length(y)

## [1] 1

hist(y)

(b) BC < 1/2

B=seq(0,1,length=2000)
C=seq(0,1,length=2000)
y=dnorm(punif(B, min = 0, max = 1) * runif(C, min = 0, max = 1))
boxplot(subset(y,y<.5),type="l", lwd=2, col="blue")

length(subset(y,y<.5))/length(y)

## [1] 1

hist(y)

(c) |B - C| < 1/2

B=seq(0,1,length=2000)
C=seq(0,1,length=2000)
y=dnorm(punif(B, min = 0, max = 1) - runif(C, min = 0, max = 1))
boxplot(subset(y,y<.5),type="l", lwd=2, col="blue")

length(subset(y,y<.5))/length(y)

## [1] 1

hist(y)

(d) max{B,C} < 1/2

B=seq(0,1,length=2000)
C=seq(0,1,length=2000)
y=dnorm(max(punif(B, min = 0, max = 1), runif(C, min = 0, max = 1)))
boxplot(subset(y,y<.5),type="l", lwd=2, col="blue")

length(subset(y,y<.5))/length(y)

## [1] 1

(e) min{B,C} < 1/2

B=seq(0,1,length=2000)
C=seq(0,1,length=2000)
y=dnorm(min(punif(B, min = 0, max = 1) , runif(C, min = 0, max = 1)))
boxplot(subset(y,y<.5),type="l", lwd=2, col="blue")

length(subset(y,y<.5))/length(y)

## [1] 1