Data 605 HW 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

1. B + C < 1/2.

n = 10000

b <- runif(n,min = 0, max = 1)

c <- runif(n,min = 0, max = 1)

a <- sum((b + c)<0.5)/n

a

## [1] 0.124

2. BC < 1/2.

sum((b*c)< 0.5)/n

## [1] 0.8464

3. |B − C| < 1/2.

sum(abs(b-c)<0.5)/n

## [1] 0.7496

4. max{B,C} < 1/2.

index <- 0

for (x in 1:n){
  if(max(c(b[x],c[x])) < 0.5 ){
    index = index + 1

}
}

index/n

## [1] 0.2438

5. min{B,C} < 1/2.

index <- 0

for (x in 1:n){
  if(min(c(b[x],c[x])) < 0.5 ){
    index = index + 1

}
}

index/n

## [1] 0.7475