Suppose $X_i$ is an indicator random variable. There is another random variable Z defined as $Z = \min(c, \sum_i X_i)$, where $c$ is a constant. How do we compute $E[Z]$? I have come up with the following expression, but I am not sure if its correct, $E[Z] = \min(c, \sum_i Pr(X_i = 1))$.
Expectation of a min function of summation of indicator random variable
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1There is no simple formula and the one you suggest is incorrect, as can be seen readily by checking some simple cases. – 2012-06-19
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0Can we atleast claim that $E[Z] \leq \min(c, \sum_i Pr(X_i = 1))$ ? – 2012-06-20
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0Yes: $Z=\min(c,S)$ implies $E(Z)\leqslant\min(c,E(S))$. – 2012-06-20