Final answer:
To find the PDF of the random variable Y, we need to find the distribution function of Y and differentiate it with respect to Y. The PDF of the random variable Y is fY(y) = 7(1-(1-y^(1/6)))⁶ * (1/6)(y^(-5/6)).
Step-by-step explanation:
To find the PDF of the random variable Y, we need to find the distribution function of Y and differentiate it with respect to Y.
Since Y=(1-X)⁶, we can find the distribution function of Y by substituting (1-X)⁶ into the given PDF of X:
fY(y) = fX(1-y^(1/6)) * (1/6)(y^(-5/6)), where 0 < y < 1.
Therefore, the PDF of the random variable Y is fY(y) = 7(1-(1-y^(1/6)))⁶ * (1/6)(y^(-5/6)).