Answer:
The answer is "0.68".
Step-by-step explanation:
Given value:
[tex]X_i \sim \frac{G_1}{2}[/tex]
[tex]E(X_i)=2 \\[/tex]
[tex]Var (X_i)= \frac{1- \frac{1}{2}}{(\frac{1}{2})^2}\\[/tex]
[tex]= \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[/tex]
Now we calculate the [tex]\bar X \sim N(2, \sqrt{\frac{2}{n}})\\[/tex]
[tex]\to \frac{\bar X - 2}{\sqrt{\frac{2}{n}}} \sim N(0, 1)\\[/tex]
[tex]\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)\\[/tex]
[tex]\to Z_n = \frac{1}{\sqrt{n}} \sum^n_{i=1} (X_i -2) \sim N(0, 2)\\[/tex]
[tex]\to P(-1 \leq X_n \leq 2) = P(Z_n \leq Z) -P(Z_n \leq -1) \\\\[/tex]
[tex]= 0.92 -0.24\\\\= 0.68[/tex]
