Final answer:
To find the expectation of the circumference and area of a rectangle with sides defined by random variables X and Y, one must calculate the expected values of these variables and then use the formulas for the expected area and expected circumference of a rectangle.
Step-by-step explanation:
The question involves finding the expectation of the circumference and area of a rectangle with sides defined by the random variables X and Y. Given that the base of the rectangle is a random variable X with a uniform distribution from 0 to 1 (X ~ U(0,1)), and the height is a random variable Y with a uniform distribution from 0 to X (Y ~ U(0, X)), we can calculate the expected values by integrating over the possible values of X and Y.
The expectation of the area, A, of the rectangle can be found by using the formula: E[A] = E[X] * E[Y], where E[X] = ½ (since it's the mean of U(0,1)) and E[Y] can be found by integrating Y from 0 to x and then integrating this result with respect to the distribution of X from 0 to 1. The expectation of the circumference, C, is given by E[C] = 2(E[X] + E[Y]), because the circumference of a rectangle is 2 times the sum of its base and height.
To solve this, we would calculate the expected value of Y when the base X is known, then integrate this function with respect to the uniform distribution of X.