How to write the PMF of a two-point distribution?

Question

I have been given the following exercise:

Let's consider the random variable $Y \sim Bern(p)$ and the stationary stochastic process:

$$(X_{t})_{t\geq0} = \begin{cases} cos(\pi t), & \mbox{if Y=1}\\ sin(\pi t), & \mbox{if Y=0} \end{cases} $$

Find the mass function of $X_{t}$.

Since the process is stationary I can consider $cos(\pi t)$ and $sin(\pi t)$ as constants. As far as I have understood the mass function of $X_{t}$ should be some kind of a generalized version of the PMF of a Bernoulli distribution, where instead of getting either 0 or 1 I get either one of $a$ or $b$. As of now I have found that this distribution is called Two-point distribution. The problem is that I cannot find its PMF.

Could you please help me? Thanks a lot in advance!

Answer 1

For a given $t\geq0$, $X_t$ may realise only two values. These change depending on $t$, but predictably.

In short, you seek $\mathsf P(X_t = \cos(\pi t))$ and $\mathsf P(X_t = \sin(\pi t))$.