The exponential pdf is a measure of lifetimes of devices that do not age. However, the exponential pdf is a special case of the Weibull distribution, which measures time to failure of devices where the probability of failure increases as time does. A Weibull random variable Y has pdf f_Y(y; α,β) = αβy^β−1e^−αyβ, 0 ≤ y, 0 < α, 0 < β.

(a) Find the maximum likelihood estimator for α assuming that β is known.

(b) Suppose α and β are both unknown. Write down the equations that would be solved simultaneously to find the maximum likelihood estimators of α and β.