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Suppose there are n i.i.d exponential random variables Yi, i=1,2...n and another random variable X independent of Yi, i=1,2...n.
The cumulative probability distribution function of X is
$$F(x)=\left\{\begin{matrix} (1-e^{-2x})^{n} &x\geq 0 \\ 0&x<0 \end{matrix}\right.$$ The probability density function of Yi is
$$f(y)=\left\{\begin{matrix} e^{-y} &y\geq 0 \\ 0&y<0 \end{matrix}\right.$$ Let S be a set of Yi s satisfying Yi < t.Thus the size of S is a binomial random variable which may not be independent of those Yi s belonging to S .
$$\left | S \right |\sim B(n,1-e^{-t})$$ Now what is the result of the probability $$ P\left ( \frac{X}{b+\sum_{j\in S}Yj}<c \right ) $$ where b and c are both positive constants? Or is it less than some value or expression?

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$Xi$ >> $X$? $ $ – Did Feb 4 '13 at 15:35
sorry,it should be X. – yyzhang Feb 5 '13 at 0:53
There is no reason to expect anything simpler than the multi-dimensional integral one is thinking about. – Did Feb 7 '13 at 13:09

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