Final answer:
The CDF of the maximum of three iid random variables with a U([0,1]) distribution is F(x) = x^3 for 0 ≤ x ≤ 1, and the corresponding PDF is f(x) = 3x^2. Thus the correct answer is (a) CDF: F(x) = x^3, PDF: f(x) = 3x^2.
Step-by-step explanation:
The student has asked to find and sketch the cumulative distribution function (CDF) and probability density function (PDF) of the random variable representing the maximum of a set of independent and identically distributed (iid) random variables with a uniform distribution U([0,1]).
When dealing with the maximum of iid uniform random variables, the CDF of the maximum, in this case max(3i), can be found by raising the CDF of a single variable to the power of the number of variables.
As all Xi have a U([0,1]) distribution, the CDF of a single Xi is F(x) = x for 0 ≤ x ≤ 1. Thus the CDF of the maximum of three such variables, max(3i), is F(x) = x3 for 0 ≤ x ≤ 1. The corresponding PDF is the derivative of the CDF, which is f(x) = 3x2, making answer (a) correct, CDF: F(x) = x3, and PDF: f(x) = 3x2.
The CDF is calculated using the formula F(x) = xn, where n is the number of random variables, and the PDF is found by differentiating the CDF with respect to x, yielding f(x) = nxn-1.