HW5

HW5

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.

Q :- Prove that B and C are proper probability distributions

# A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. This range will be between the minimum and maximum statistically possible values.

n <- 10000
B <- as.data.frame(runif(n, min=0, max=1))
C <- as.data.frame(runif(n, min=0, max=1))



min(B)

## [1] 0.0001683552

max(B)

## [1] 0.9998546

# checking the skewness of B
hist(B[,1], prob=TRUE, ylim=c(0,1.5), main = 'Histogram of B', xlab = '')

min(C)

## [1] 5.254173e-05

max(C)

## [1] 0.9999642

#checking skewness of C
hist(C[,1], prob=TRUE, ylim=c(0,1.5), main = 'Histogram of B', xlab = '')

j = 0
for(i in 1:length(B)){
  if(B[i]+C[i] < 0.5){
    j = j+1
  }
}

## Warning in if (B[i] + C[i] < 0.5) {: the condition has length > 1 and only
## the first element will be used

print(paste("The Probabilty B+C < 1/2 =", j/length(B)))

## [1] "The Probabilty B+C < 1/2 = 0"

As we can see bins of both the histogram are evenly distributed, B & C is proper probability distributions.

a <- sum((B+C) < .5)/n
print(paste("The probability of B+C less than 1/2 is",a))

## [1] "The probability of B+C less than 1/2 is 0.1244"

b <- (sum((B*C) < .5)/n)
print(paste("The probability of B*C less than 1/2 is",b))

## [1] "The probability of B*C less than 1/2 is 0.8483"

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

## [1] "The probability of |B-C| be less than 1/2 is 0.7362"

d <- 1
B <- runif(n, min=0, max=1)
C <- runif(n, min=0, max=1)
for(i in 1:length(B)){
  if(max(B[i], C[i]) < 0.5){
    d = d + 1
  }
}
d <- d/10000
print(paste("The probability of max{B,C} less than 1/2 is",d))

## [1] "The probability of max{B,C} less than 1/2 is 0.253"

e <- 1
B <- runif(n, min=0, max=1)
C <- runif(n, min=0, max=1)
for(i in 1:length(B)){
  if(min(B[i], C[i]) < 0.5){
    e = e + 1
  }
}
e <- e/10000
print(paste("The probability of min{B,C} less than 1/2 is",d))

## [1] "The probability of min{B,C} less than 1/2 is 0.253"