Question

For i≥1 , let Xi∼G1/2 be distributed Geometrically with parameter 1/2 . Define Yn=1n−−√∑i=1n(Xi−2) Approximate P(−1≤Yn≤2) with large enough n .

Answer 1

The answer is "0.68".

Explanation:

Given value:

`X_i \sim (G_1)/(2)`

`E(X_i)=2 \\`

`Var (X_i)= (1- (1)/(2))/(((1)/(2))^2)\\`

`= ( (2-1)/(2))/((1)/(4))\\\\= ( (1)/(2))/((1)/(4))\\\\= (1)/(2) * (4)/(1)\\\\= (4)/(2)\\\\=2`

Now we calculate the
`\bar X \sim N(2, \sqrt{(2)/(n)})\\`

`\to \frac{\bar X - 2}{\sqrt{(2)/(n)}} \sim N(0, 1)\\`

`\to \sum^n_(i=1) (X_i - 2)/(n) *\sqrt{(n)/(2)}} \sim N(0, 1)\\\\\to \sum^n_(i=1) (X_i - 2)/(√(2n)) \sim N(0, 1)\\`

`\to Z_n = (1)/(√(n)) \sum^n_(i=1) (X_i -2) \sim N(0, 2)\\`

`\to P(-1 \leq X_n \leq 2) = P(Z_n \leq Z) -P(Z_n \leq -1) \\\\`

`= 0.92 -0.24\\\\= 0.68`