Question_1

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

library(ggplot2)
set.seed(10000)
B <- (B = runif(1000, min = 0, max = 1))
C <- (C = runif(1000, min = 0, max = 1))
ecdf(B)
## Empirical CDF 
## Call: ecdf(B)
##  x[1:1000] = 0.00044016, 0.0011836, 0.0013299,  ..., 0.99956, 0.99966
ecdf(C)
## Empirical CDF 
## Call: ecdf(C)
##  x[1:1000] = 0.0016067, 0.0020639, 0.0020831,  ..., 0.99961, 0.99964

Both variables have a probability within [0-1] interval

summation = sum(punif((B+C)<0.5, min=0, max=1)) / 1000
summation
## [1] 0.111

(b)

BC < 0.5

product = sum(punif((B*C)<0.5, min=0, max=1)) / 1000
product
## [1] 0.843

(c)

|B − C| < 1/2

subtrac = sum(punif(abs(B-C)<0.5, min=0, max=1)) / 1000
subtrac
## [1] 0.713

(d)

max{B,C} < 1/2.

max_sum = sum((pmax(B,C)) < 0.5)/1000
max_sum
## [1] 0.21

(e)

min{B,C} < 1/2

min_sum = sum((pmin(B,C)) < 0.5)/1000
min_sum
## [1] 0.745