Final answer:
The question requires using integral calculus to find the mean and variance of a random variable with a given probability density function, by integrating with respect to that function over its support.
Step-by-step explanation:
The question concerns finding the mean and variance of a random variable y with a given density function f(y). To find the mean (also called the expected value) of y, one would integrate the product of y and f(y) over all possible values of y. The variance of y is found by integrating the square of the difference between y and its mean, multiplied by f(y), over all possible values of y. This process requires knowledge of integral calculus and the properties of probability density functions.
For example, if f(y) = y, the mean would be calculated as ∫ y * f(y) dy, with the limits of integration being the support of y (where f(y) is non-zero). The variance would be calculated as ∫ (y - μ)2 * f(y) dy where μ is the mean of y.
In cases where specific functions were not provided, the question cannot be answered numerically but only the procedure can be described. The answer would also depend on whether the density function is properly normalized to ensure that the total probability is one.