Question

What is the value of E[Xi²] in terms of μ and σ² for independent, identically distributed random variables X¹, X², …, Xⁿ with mean μ and variance σ²? Please provide a numerical confirmation using X¹ ~ gamma (shape=3, scale=2), which has a mean of 6 and a variance of 12.

Answer 1

The value of E[Xi²] for IID random variables X with mean μ and variance σ² is μ² + σ². For X¹ ~ gamma (shape=3, scale=2), which has a mean of 6 and a variance of 12, E[Xi²] = 6² + 12 = 48.

Step-by-step explanation:

The value of E[Xi²] in terms of μ and σ² for independent, identically distributed random variables X¹, X², …, Xⁿ with mean μ and variance σ² is given by

E[Xi²] = μ² + σ².

To provide a numerical confirmation using X¹ ~ gamma (shape=3, scale=2),

we can find the mean and variance of this distribution. In this case, the mean μ is 6 and the variance σ² is 12.

Plugging these values into the formula, we get

E[Xi²] = 6² + 12 = 48.