91影视

a gamma random variable with parameters$(t,位)$has an approximately normal distribution when$t$

is large.

Suppose that$X$has a gamma distribution with parameters$(t,位),t>0,位>0$. Then, this random variable can be represented as

$X=X_1+X_2+鈰�+X_t,$

whereby the random variables$X_i$are independently identically distributed. More precisely, for each$i=1,2鈥�,t$, the variable $X_i$is an exponential random variable with a parameter$位$. Therefore, for each$i=1,2鈥�,t$, the variable$X_i$has mean

$渭=E\left[X\right]=\frac{1}{位}$

and variance

${蟽}^2=Var\left(X\right)=\frac{1}{{位}^2}$

The central limit theorem says that the distribution of

$\frac{X-t渭}{蟽\sqrt{t}}$

tends to the standard normal distribution $t鈫�鈭�$i.e. for eachrole="math" localid="1649862027173" $a鈭�\mathrm{鈩�}$

$P\left\{\frac{X-\frac{t}{位}}{\frac{\sqrt{t}}{位}}鈮�a\right\}鈫�桅\left(a\right)$

Hence, for large$t,X$has an approximately normal distribution with mean $\frac{t}{位}$and variance$\frac{t}{{位}^2}$.