Question

Suppose you have a set of independent random variables X1, X2, ..., Xn. Each random variable X_i follows a gamma distribution with the probability density function (pdf) as given:

f(x | x0, α) = (Γ(α)^-1) * x^(α-1) * e^(-x) for x ≥ x0, and 0 otherwise.

Here, x0 is given as a constant, but α is unknown. You are interested in finding the Maximum Likelihood Estimator (MLE) of Γ(α) / Γ'(α).

Formulate a question to find this MLE for Γ(α) / Γ'(α).

Answer 1

To find the Maximum Likelihood Estimator (MLE) of Γ(α) / Γ'(α) for the given gamma distribution with X_i as independent random variables, you need to solve the likelihood function using differentiation and substitution.

Step-by-step explanation:

To find the Maximum Likelihood Estimator (MLE) of Γ(α) / Γ'(α) for the given gamma distribution, consider the likelihood function L(α) = Π_{i=1}^{n} f(X_i) = Π_{i=1}^{n} ((Γ(α)^-1) * x_i^(α-1) * e^(-x_i)).

To simplify the likelihood function, take the logarithm of L(α) and differentiate it with respect to α. Set the derivative equal to zero and solve for α to obtain the MLE.

Finally, substitute the obtained value of α into Γ(α) / Γ'(α) to find the MLE of Γ(α) / Γ'(α).