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

Answer:

The answer is "0.68".

Step-by-step explanation:

Given value:

$X_i \sim \frac{G_1}{2}$

$E(X_i)=2$

$Var (X_i)= \frac{1- \frac{1}{2}}{(\frac{1}{2})^2}$

             $= \frac{ \frac{2-1}{2}}{\frac{1}{4}}\\\\= \frac{ \frac{1}{2}}{\frac{1}{4}}\\\\= \frac{1}{2} \times \frac{4}{1}\\\\= \frac{4}{2}\\\\=2$

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

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

$\to \sum^n_{i=1} \frac{X_i - 2}{n} \times\sqrt{\frac{n}{2}}} \sim N(0, 1)\\\\\to \sum^n_{i=1} \frac{X_i - 2}{\sqrt{2n}} \sim N(0, 1)$

$\to Z_n = \frac{1}{\sqrt{n}} \sum^n_{i=1} (X_i -2) \sim N(0, 2)$

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

                               $= 0.92 -0.24\\\\= 0.68$