Please help me with this small probability problem.
Let U be uniformly distributed over the intervall [0, 2π], calculate
E[cos(U)] {that is the expected value for another stochastic variable equal to cos(U)}
E[sin(U)^2]
My attempt:
Let f(x) be the probability density function for U, it's obvious that
f(x) = { 1/(2π) | 0 < x < 2π
0 | else }
and the cumulative distribution function for U
F(x) = df/dx = { x/(2π) | 0 < x < 2π
0 | else }
My problem is, I don't know how to calculate a density function for cos(U), because cos(U) is not strictly growing in [0, π]