Which property of the moment generating function allows us to uniquely determine the distribution of a random variable?

What is the moment generating function (MGF) of a random variable X?

Explain the concept of moment generating function. What are its limitations.

Which of the following statements about the moment generating function (MGF) is true?Group of answer choicesThe MGF is unique to each random variable.The MGF is a probability density function.The MGF always exists for discrete random variables.The MGF is undefined for continuous random variables.

Write down the principle of moments

Suppose X = (X1, . . . , Xn) is a random sample from the Gamma distribution withshape α and rate θ, i.e.,fX (x) = θαΓ(α)xα−1e−xθ, x > 0, α > 0, θ > 0.Each Xi has expectation E(X) = α/θ, variance Var(X) = α/θ2, and momentgenerating function MX (t) = (1 − t/θ)−α.(a) Assuming both α and θ to be unknown, write down the log likelihood functionℓ(θ, α; X) and the corresponding score functions ∂ℓ∂θ and ∂ℓ∂α .(b) Verify that each score function has zero expectation.(c) Assuming α to be known:(i) Calculate the Cramer Rao Lower Bound (CRLB) for the variance of anunbiased estimator of θ.(ii) Is there any unbiased estimator of θ whose variance attain the CRLB?(iii) Show thatS = α − 1nnXi=1 1Xiis an unbiased estimator for θ. What is the MVU estimator for θ?(iv) Identify a change of parameter η = η(θ) for which there exists an unbiasedestimator with variance attained the CRLB.(v) For the parameter in the previous part, identify the MVU estimator whosevariance attains the CRLB. Compute this variance