Law of the unconscious statistician

Problem: Let X be a uniform random variable on [0,1], and let $Y=\tan(\pi(x - 1/2))$. Calculate E(Y) if it exists.

Note that the expected value formula ∫ t*f(t) dt does not apply in this case. If we use that, we are computing the expected value of X rather than Y. Then how can we find E(Y)? We use Law of the unconscious statistician (LOTUS):

The law of the unconscious statistician is a theorem used to calculate the expected value of a function g(X) of a random variable X when one knows the probability distribution of X but one does not explicitly know the distribution of g(X).

The expected value of Y = g(X) where $g(x)=\tan(\pi(x - 1/2))$ is given as follows:

$\int_{-\infty}^{\infty} g(x) p(x) dx$

where p(x) is the probability density function of X. Here p(x) is 1 for $0 \leq x \leq 1$ and 0 otherwise.

Hence we want to evaluate $\int_0^1 \tan(\pi(x - 1/2)) dx$, which is $-\pi^{-1}\ln \cos(\pi(x-1/2))|_0^1$. So it is infinity on either (0, 1/2) or (1/2, 1). Thus, the expected value of Y does not exist.

Intuitive explanation: At each value of x, there's a probability of p(x) of having that value, and the corresponding value of Y is just g(x), so add up g(x) p(x) over all x to get the expected value of Y.

More to explore:
http://www.quora.com/What-is-an-intuitive-explanation-of-the-Law-of-the-unconscious-statistician