Question

Let X 1 ,X 2 ,…,X n be a random sample from a RVX∼Gamma(α,β). Let Kn=Xˉ −αβ/β√α/n ​ . compute the first three moments of K n and compute it with first three moments of the standard normal distribution.

Answer 1

The mean of Kn is given by αβ - αβ/√(αn), the variance is αβ², and the skewness depends on the skewness of X. The moments of Kn can be compared with the moments of the standard normal distribution.

Step-by-step explanation:

To compute the first three moments of Kn, we need to find its mean, variance, and skewness. Let's start by finding the mean:

Using the properties of the gamma distribution, we know that the mean of X is αβ, so the mean of Kn is given by:

E(Kn) = E(X - αβ/β√(α/n))

= E(X) - αβ/β√(α/n)

= αβ - αβ/β√(α/n)

= αβ - αβ/√(αn)

Next, let's find the variance:

Var(Kn) = Var(X - αβ/β√(α/n))

= Var(X)

= αβ²

Finally, let's find the skewness:

To find the skewness, we need the third central moment (E[(X - μ)³]).

However, since the gamma distribution does not have a simple formula for its higher moments, we can't directly find the skewness of Kn.

To compare it with the skewness of the standard normal distribution, we can use the fact that the skewness of a sum of independent random variables is the sum of their individual skewness values.

Since Kn is the difference between two random variables (X and αβ/β√(α/n) which does not depend on X), its skewness will depend only on the skewness of X.

The skewness of X is given by:

Skew(X) = (2√α)/β

So, to compare the moments of Kn with the moments of the standard normal distribution, we can compare their means, variances, and skewness values.

Hence, the mean of Kn is given by αβ - αβ/√(αn), the variance is αβ², and the skewness depends on the skewness of X.