91Ó°ÊÓ

Given$X$is a gamma random variable with parameters$(n,1)$,

Assume that$X$has a gamma distribution with parameters $\mathrm{Î\pm }=n$and$\mathrm{λ}=1$. Therefore, the variable$X$is a sum of $n$independent variables $X_i$:

$X=X_1+X_2+⋯+X_n$

whereby each random variable has an exponential distribution with parameters$\mathrm{λ}$. The mean and the variance of variables $X$are

$E\left[X\right]=\frac{\mathrm{Î\pm }}{\mathrm{λ}}=\frac{n}{1}=n$

and

$Var\left(X\right)=\frac{\mathrm{Î\pm }}{{\mathrm{λ}}^2}=\frac{n}{1^2}=n$.

The central limit theorem says that the average of a set of independent identically distributed random variables is approximately normally distributed

for each$a$,

$P\left\{\frac{X-E\left[X\right]}{\sqrt{Var\left(X\right)}}≤a\right\}→\mathrm{Φ}\left(a\right)$

Now, using this theorem we get:

$$\begin{gathered} .01>P\left\{\left|\frac{X}{n}-1\right|>.01\right\}=1-P\left\{-.01≤\frac{X}{n}-1≤.01\right\}= \\ 1-P\left\{-.01\sqrt{n}≤\frac{X-n}{\sqrt{n}}≤.01\sqrt{n}\right\}\stackrel{(*)}{≈}1-\left[\mathrm{Φ}\right(.01\sqrt{n})-\mathrm{Φ}(-.01\sqrt{n}\left)\right] \\ \mathrm{Φ}(-z)=1-\mathrm{Φ}\left(z\right)=1-\left[2\mathrm{Φ}\right(.01\sqrt{n})-1]⇒\mathrm{Φ}(.01\sqrt{n})>.995 \end{gathered}$$

$\stackrel{\text{Table}5.1\text{(textbook, Chapter 5)}}{⇒}.01\sqrt{n}>2.58⇒\sqrt{n}>258⇒n>{258}^2=66564$