Suppose I have $n$ observations of binary data $(X, Y, Z, \ldots)$ and suppose I want to estimate $\theta = P(X = 1 | Y = 1)$. It seems clear to me that

$$\frac{\sum_{i}\mathbb{1}_{X_i = 1, Y_i = 1}}{\sum_i\mathbb{1}_{Y_i = 1}}$$

is a good estimator for $\theta$. However, I want to show some properties of this estimator (like unbiasedness/variance) but I can't figure out how to easily take the expectation since both the numerator and denominator are random and dependent. Thanks!



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