Question

Find the expectation of (a) a constant random variable X, X(w) =a for all w (b) X : [0, 1] → R given by X(w) = min{w, 1-w} (the distance to the nearest endpoint of the interval [0, 1]) (c) X : [0, 1]² → R, the distance to the nearest edge of the square [0, 1]².

Answer 1

The expectation of a constant random variable, a random variable representing the distance to the nearest endpoint of an interval, and a random variable representing the distance to the nearest edge of a square.

Step-by-step explanation:

a. The random variable X is a constant random variable, where X(w) = a for all w. In other words, no matter what the outcome is, X will always be equal to the constant value a.

b. The random variable X is defined as the minimum of w and 1-w, which represents the distance to the nearest endpoint of the interval [0, 1]. For example, if w is 0.3, then X will be 0.3, as it is closer to the endpoint 0.

c. The random variable X represents the distance to the nearest edge of the square [0, 1]². This means that X(w) will be the minimum of the distance from w to the nearest vertical edge and the distance from w to the nearest horizontal edge.