YMai_605_Assignment5

Introdcution to Probability

Chapter 1 Discrete Probability Distributions

Chapter 2 Continuous Probability Densities

Questions:

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

1. B + C < 1/2.

2. BC < 1/2.

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

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

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

Solutions:

Prove that B and C are proper probability distributions.

Proof.

# generate random numbers from a uniform distribution
set.seed(NULL)
B <- runif(10000, min = 0, max = 1)  

set.seed(NULL)
C <- runif(10000, min = 0, max = 1)   

df_b<- as.data.frame(B)

# check whether the probability of B is between 0 and 1

# firsr create a series with all the values as False
sum_p <- rep(F,times=length(B))

# then calculate the pribability of B
for (i in 1:length(B)){
  x=1
  for (j in 1:length(B)){
    if (j != i && B[i] == B[j]){
      x <- x + 1
    }
  }
  p <- x/length(B)
  # if the probability is between and 1 inclusive, the value in sum_p will change to True correspondingly 
  sum_p[i] <- (0 <= p && p<= 1) 
}

condition_1 <- !any(sum_p == F)

print(paste("Is all the probabilities of B is between 0 and 1 inclusive? ",condition_1 ))

## [1] "Is all the probabilities of B is between 0 and 1 inclusive?  TRUE"

print(paste("Similarly, all the probabilities of B is between 0 and 1 inclusive "))

## [1] "Similarly, all the probabilities of B is between 0 and 1 inclusive "

# sum the probability of B and denote as sigma
sigma_b <- 0
for (i in 1:length(B)){
  x=1
  for (j in 1:length(B)){
    if (j != i && B[i] == B[j]){
      x <- x + 1
    }
  }
  p <- x/length(B)
  sigma_b <- sigma_b + p 
}

sigma_b

## [1] 1

sigma_b <- round(sigma_b)

print(paste("The sum of the probabilities of the outcomes of B is",sigma_b))

## [1] "The sum of the probabilities of the outcomes of B is 1"

print(paste("Similarly,The sum of the probabilities of the outcomes of C is",sigma_b))

## [1] "Similarly,The sum of the probabilities of the outcomes of C is 1"

For both B or C, all the probabilities is between 0 and 1 inclusive.

For both B or C,the sum of the probabilities of the outcomes is 1.

So B and C are proper probability distributions

(a) B + C < 1/2.

P1 <- sum((B+C) < 1/2)/length(B)

(paste("The probability for B + C < 1/2. is: ",P1))

## [1] "The probability for B + C < 1/2. is:  0.1284"

(b) BC < 1/2.

P2 <- sum((B*C) < 1/2)/length(B)
(paste("The probability for BC < 1/2. is: ",P2))

## [1] "The probability for BC < 1/2. is:  0.8497"

(c) |B - C| < 1/2.

P3 <- sum( abs((B-C)) < 1/2)/length(B)
(paste("The probability for |B - C| < 1/2. is: ",P3))

## [1] "The probability for |B - C| < 1/2. is:  0.7495"

x=0
for (i in 1:length(B)){
  if (max(B[i],C[i]) < 1/2){
    x=x+1
  }
}

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

P4 <- x/length(B)
(paste("The probability for max{B,C} < 1/2. is: ",P4))

## [1] "The probability for max{B,C} < 1/2. is:  0.2522"

y=0
for (i in 1:length(B)){
  if (min(B[i],C[i]) < 1/2){
    y=y+1
  }
}

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

P5 <- y/length(B)
(paste("The probability for min{B,C} < 1/2. is: ",P5))

## [1] "The probability for min{B,C} < 1/2. is:  0.7523"