Data605 Week8 Disscusion

13 Suppose one hundred numbers X1, X2, . . . , X100 are chosen independently at random from [0, 20]. Let S = X1 + X2 + · · · + X100 be the sum, A = S/100 the average, and \(S^* = (S-1000)/(10/ \sqrt3)\) the standardized sum.

Find lower bounds for the probabilities.

  1. \(P(|S-1000|\le 100).\)
set.seed(123)
S<-replicate(1000, {
              s<-sum(runif(100,0,20))-1000})
sum(abs(S)<=100)/1000
## [1] 0.925

(Since the Xn has unif distribution with mean =10, base on law of large number: \(P(|S-1000|\le 100).\) =1 when n approximate to infinity. I will add this part later.)

  1. \(P(|A-10|\le 1)\)
set.seed(123)
a<-replicate(1000, {
              A<-(sum(runif(100,0,20))/100) - 10})
sum(abs(a)<=1)/1000
## [1] 0.925

(Since the Xn has unif distribution with mean =10, base on law of large number: \(P(|A-10|\le 1)\) =1 when n approximate to infinity. I will add this part later.)

  1. \(P(|{ S }^{ * }|\le \sqrt { 3 } ).\)
set.seed(123)
s_star<-replicate(1000, {
              S_star<-(sum(runif(100,0,20))-1000)/(10/sqrt(3))})
sum(abs(s_star)<=sqrt(3))/1000
## [1] 0.126

(Since the Xn has unif distribution with mean =10, base on law of large number: \(P(|{ S }^{ * }|\le \sqrt { 3 } ).\) = 0.1 when n approximate to infinity. I will add this part later.)