problem set 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.

Finding probability:

B <- runif(10000, min = 0, max = 1)
C <- runif(10000, min = 0, max = 1)
hist(B, col = "yellow")

hist(C, col = "orange")

(a) B + C < 1/2.

X <- B + C
hist(X, main = "Histogram of B + C", col = "skyblue")

(sum(X < 0.5) / 10000)
## [1] 0.1205

(b) BC < 1/2

X <- B * C
hist(X, main = "Histogram of B x C", col = "skyblue")

(sum(X < 0.5) / 10000)
## [1] 0.8432

(c) |B − C| < 1/2.

X <- abs(B - C)
hist(X, main = "Histogram of |B − C|", col = "skyblue")

(sum(X < 0.5) / 10000)
## [1] 0.7474

(d) max{B,C} < 1/2.

X <- pmax(B,C)
hist(X, main = "Histogram of max{B,C}", col = "skyblue")

(sum(X < 0.5) / 10000)
## [1] 0.2483

(e) min{B,C} < 1/2.

X <- pmin(B,C)
hist(X, main = "Histogram of min{B,C}", col = "skyblue")

(sum(X < 0.5) / 10000)
## [1] 0.7474