Suppose that X1,X2,...,X200 is a set of independent and identically distributed Gamma random variables with parameters α=4,λ=3. Describe the normal distribution that would correspond to the sum of those 200 random variables if the Central Limit Theorem holds.

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Suppose that X1,X2,...,X200 is a set of independent and identically distributed Gamma random variables with parameters α=4,λ=3.   Describe the normal distribution that would correspond to the sum of those 200 random variables if the Central Limit Theorem holds.

au4gsf · Accepted Answer

Step 1 We have been given X1,X2,X3,…………..X200 are independently and identically distributed gamma random variables with prop =4 and λ=3. Thus, we have the mean and variance for gamma distribution is given as. Mean=μ=αλ=4×3=12 Variance=σ2=αλ2=4×32=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=X1+X2+X3+……..X200 Mean(X)=E(X1+X2+X3+........X200) =E(X1)+E(X2)+......E(X200) =12×200=2400 Step 3 Variance(X)=V(X1+X2+X3+.....X200) =V(X1)+V(X2)+....V(X200) (since X1X2.....X200 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∼N(2400,7200)

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.

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