Assignnment 5

b<- runif(1000, min = 0, max =1)
c <-runif(1000, min = 0, max =1)
d = b + c
e = b*c
f = abs(b - c)
g = max(b)
h = max(c)
i = min(b)
j = min(c)

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

k = pnorm(.5,mean(d),sd(d))
k

## [1] 0.1121062

2. BC < 1/2

l=pnorm(.5,mean(e),sd(e))
l

## [1] 0.8751556

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

m= pnorm(.5,mean(f),sd(f))
m

## [1] 0.7331875

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

n = pnorm(.5,mean(b),sd(b))
o =pnorm(.5,mean(c),sd(c))
m*o

## [1] 0.3598815

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

p = 1 -pnorm(.5,mean(b),sd(b))
q =1 - pnorm(.5,mean(c),sd(c))
p*q

## [1] 0.2463206

```