VStoyanova_Assign5

HW5

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.

All the probabilities must be between 0 and 1 inclusive The sum of the probabilities of the outcomes must be 1.

b <- runif(n = 1000000,0,1)
c <- runif(n = 1000000,0,1)

hist(b)

hist(c)

B and C are between 0 and 1 inclusive and the sum of the probabilities of the outcomes must be 1 therefore 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.

P1 <-sum((b+c) < 1/2)/length(c)
P1

## [1] 0.124838

2. BC < 1/2.

P2 <-sum((b*c) < 1/2)/length(b)
P2

## [1] 0.846191

3. |B - C| < 1/2.

P3 <-sum( abs((b-c)) < 1/2)/length(b)
P3

## [1] 0.750129

By simulation:

s1<- 0
for (i in 1:length(b)) {
  if (abs(b[i]-c[i]) < 1/2) {
    s1 <- s1 + 1
  } 
}
P3<- s1 / length(b)
P3

## [1] 0.750129

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

s2<- 0
for (i in 1:length(b)) {
  if (max(b[i],c[i]) < 1/2) {
    s2 <- s2 + 1
  } 
}
P4<- s2 / length(b)
P4

## [1] 0.249978

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

s3 <- 0
for (i in 1:length(b)) {
  if (min(b[i],c[i]) < 1/2) {
    s3 <- s3 + 1
  } 
}
P5<- s3/ length(b)
P5

## [1] 0.750156