 Emily-Jane Bray

2021-02-26

Suppose that ${X}_{1},{X}_{2},...,{X}_{200}$ is a set of independent and identically distributed Gamma random variables with parameters $\alpha =4,\lambda =3$.
Describe the normal distribution that would correspond to the sum of those 200 random variables if the Central Limit Theorem holds. au4gsf

Step 1
We have been given ${X}_{1},{X}_{2},{X}_{3},\dots \dots \dots \dots ..{X}_{200}$ are independently and identically distributed gamma random variables with prop .
Thus, we have the mean and variance for gamma distribution is given as.
$Mean=\mu =\alpha \lambda =4×3=12$
$Variance={\sigma }^{2}=\alpha {\lambda }^{2}=4×{3}^{2}=36$.
Step 2
Since, the random sample is quite large enough of size 200 and also the random variables are independently and identically distributed.
Hence, according to the central limit theorem, the sum of these random variables has normal distribution with mean and standard deviation which is given as below.
Where, $X={X}_{1}+{X}_{2}+{X}_{3}+\dots \dots ..{X}_{200}$
$Mean\left(X\right)=E\left({X}_{1}+{X}_{2}+{X}_{3}+........{X}_{200}\right)$
$=E\left({X}_{1}\right)+E\left({X}_{2}\right)+......E\left({X}_{200}\right)$
$=12×200=2400$
Step 3
$Variance\left(X\right)=V\left({X}_{1}+{X}_{2}+{X}_{3}+.....{X}_{200}\right)$
$=V\left({X}_{1}\right)+V\left({X}_{2}\right)+....V\left({X}_{200}\right)$ (since ${X}_{1}{X}_{2}.....{X}_{200}$ are independently distributed)
$=36×200=7200$
Thus, the normal distribution that would correspond to the sum of those 200 random variables if the Central Limit Theorem holds is
$X\sim N\left(2400,7200\right)$

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