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  • $X$ 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.

R zu
  • 687
  • seems related to https://math.stackexchange.com/questions/610443/finding-a-correlation-between-bernoulli-variables – R zu Dec 12 '18 at 17:00
  • Can't do much in this case. Have to assume that the two probability variables are independent at each point of time. – R zu Dec 12 '18 at 18:39
  • How do you get $x_t$ and $y_t$? Are they given? Are ${(X_t,Y_t)}$ i.i.d? –  Dec 12 '18 at 19:24
  • Something here doesn't make too much sense; what is $(X_t,Y_t)$? It seems like you want $(X_t,Y_t)_{t}$ to be independent samples of $(X,Y)$, but their parameters seem to be $(x_t,y_t)$, which further seem to be changing. – Marcus M Dec 12 '18 at 19:25

1 Answers1

1

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

hunter
  • 32,629