Final answer:
To find the distribution of W = Y^2 where Y follows a Beta distribution, both the Method of Transformations and the Method of Distribution Functions can be used and will yield the same result when applied correctly.
Step-by-step explanation:
The question pertains to finding the distribution of a random variable W, which is a transformation of another random variable Y that follows a Beta distribution with parameters α and β. Specifically, W is defined as W = Y2. To find the distribution of W, we have two main methods: the Method of Transformations and the Method of Distribution Functions. In this case, the correct answer is D) Both methods will work, and they will give the same answer.
Method of Transformations
The Method of Transformations involves using the probability density function (PDF) of Y to derive the PDF of W. This method leverages the known behavior of Y to establish how W behaves. The process includes finding the distribution of W by computing the Jacobian of the transformation and applying it to the PDF of Y.
Method of Distribution Functions
The Method of Distribution Functions relies on working with the cumulative distribution function (CDF) of Y to find the CDF of W and then differentiating it to get the PDF of W. It starts with FW(w) = P(W ≤ w), calculating the corresponding values of Y that satisfy W = Y2, and then transforming this into the CDF of W, from which we can obtain the PDF by differentiation.
Both methods require careful calculations but will result in the same probability distribution for W when applied correctly. This is because both methods are simply different approaches to describing the same probabilistic transformation.