Estimate relationship between two Bernoulli random variables

Question

• $X$ and $Y$ are Bernoulli random variables
• $X$ and $Y$ are not independent
• $x_{t} = P(X_t = 1)$ and $y_{t} = P(X_t = 1)$ for time $t$.
• Is it possible to estimate $P(Y = 1 | X = 1)$ from many pairs of $x_{t}$ and $y_{t}$?

I tried:

$$\frac{\sum_{t}y_{t}x_{t}}{\sum_{t}x_{t}}$$

But this formula doesn't make sense because that means $P(X = 1|X = 1)$ is

$$\frac{\sum_{t}x_{t}x_{t}}{\sum_{t}x_{t}}$$

which is not necessarily 1.

Answer 1

Throw away the data where $x_t \neq 1$. Then compute the proportion of the remaining data where $y_t = 1$.