Assignment 5

Probabilities Assignment

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:

B + C < 0.5

set.seed(5) 
n=10000
B = runif(n, min = 0, max = 1)
set.seed(6)
C = runif(n, min = 0, max = 1)

sum((B+C)<0.5)/n

## [1] 0.1262

The probability of a value sum being less than 0.5 is 12.6%.

B*C < 0.5

sum((B*C)<0.5)/n

## [1] 0.8375

The probability of a value product being less than 0.5 is 84%.

|B - C| < 0.5

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

## [1] 0.7499

The probability of a value being less than 0.5 is 0.75%.

max(B,C) < 0.5

count <- 0
for(i in 1:n){
  if(max(c(B[i],C[i])) < 0.5){
    count = count+1
  }
}
count/n

## [1] 0.2488

The probability of max(B,C) being less than 0.5 is 0.25%

min(B,C) < 0.5

count <- 0
for(i in 1:n){
  if(min(c(B[i],C[i])) < 0.5){
    count = count+1
  }
}
count/n

## [1] 0.7483

The probability of min(B,C) being less than 0.5 is 0.75%