Data 605 HW 5

PROVE THAT B & C ARE PROPER PROBAILITY DISTRUBUTIONS

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.

Random number selection

B <- runif(10000, min = 0, max = 1)

C <- runif(10000, min = 0, max = 1)

head(B)

## [1] 0.2452038 0.6086263 0.9209765 0.9175817 0.6109816 0.3463886

head(C)

## [1] 0.239169553 0.464629650 0.145858699 0.001651942 0.145754649 0.255741605

head(B+C)

## [1] 0.4843733 1.0732559 1.0668352 0.9192336 0.7567363 0.6021302

After creating a random sample space, the values of B and C fall between 0 and 1 and the sum of their probabilites, B and C, will equal 1.

B+C < 1/2

x <- sum((B+C) < .5)/10000

print(paste("The probability of B+C being less than 1/2 is",x))

## [1] "The probability of B+C being less than 1/2 is 0.1169"

BC < 1/2

y <- (sum((B*C) < .5)/10000)
print(paste("The probability of BC being less than 1/2 is",y))

## [1] "The probability of BC being less than 1/2 is 0.8432"

|B-C| < 1/2

w <- sum(abs((B-C)) < .5)/10000
print(paste("The probability of |B-C| being less than 1/2 is",w))

## [1] "The probability of |B-C| being less than 1/2 is 0.7473"

MAX{B,C} < 1/2

x <- 0
for(i in 1:10000){
  if(max(c(B[i],C[i])) < 0.5){
    x = x+1
  }
}
y <- x/10000
print(paste("The probability that max{B,C} will be less than 1/2 is",y))

## [1] "The probability that max{B,C} will be less than 1/2 is 0.2487"

MIN{B,C} < 1/2

z <- 0
for(j in 1:10000){
  if(min(c(B[j],C[j])) < 0.5){
    z = z+1
  }
}
w <- x/10000
print(paste("The probability that min{B,C} will be less than 1/2 is",w))

## [1] "The probability that min{B,C} will be less than 1/2 is 0.2487"