The document discusses the properties of a random variable x that follows a uniform distribution on the interval (a,b). It demonstrates that the expected value epsilon(x) is the midpoint of the interval, calculated as (a + b)/2. The derivation involves integrating the probability density function f(x) and simplifying the result.

1. Suppose that the random variable X has the uniform distribution on the interval (a,b). Show that
epsilon(X) = (a + b)/2
Solution
since it's a uniform distribution f(x)=1/(b-a).E(x)= (int_{a}^{b}xf(x) dx) => x^2/2*(b-a)|a,b
=>(b^2-a^2)/2*(b-a) => b+a/2