Question

The Gamma distribution is often used to describe the wait time until a certain number of events occurs. Assume x₁, ....., xₙ ∼ ᵢᵢ Gamma(α,β), where α is shape and β is rate. Calculate the maximum likelihood estimator of β either by hand The probability density function (PDF) for the Gamma distribution is: f(x∣α,β)= βα / Γ (α) xα⁻¹ e⁻βˣ

Answer 1

To calculate the maximum likelihood estimator of beta in the Gamma distribution, we need to find the value of beta that maximizes the likelihood function.

Step-by-step explanation:

To calculate the maximum likelihood estimator of beta in the Gamma distribution, we need to find the value of beta that maximizes the likelihood function. The likelihood function is the product of the probability density function (PDF) values for each observation.

The PDF for the Gamma distribution is:

f(x|alpha,beta) = (beta^alpha / Gamma(alpha)) * x^(alpha-1) * e^(-beta*x)

We can take the natural logarithm of the likelihood function to simplify calculations. Then we take the derivative of the logarithm of the likelihood function with respect to beta and set it equal to zero to find the maximum likelihood estimator of beta.