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I have some set of known size but with unknown elements, $(x_1, ..., x_N) \in X$, where the elements of $X$ are exponentially distributed random variables with unknown rate parameters, $(\lambda_1, ..., \lambda_N) \in R$. I also have a "black box" function $f$ that samples an element from $X$ with uniform probability, and then returns a randomly sampled value from the chosen element's exponential distribution (corresponding, perhaps, to the time until the first instance of an event governed by the chosen variable).

I'm looking to use $f$ to discern whether or not an exponentially distributed random variable, $x_q$, with known rate parameter, $\lambda_q$, exists in the set $X$. I also know that $\lambda_q$ is smaller then all other rate parameters in the set $X$ by at least a multiplicative factor $w$. Said another way, $\lambda_q \leq Min[(R-\lambda_q)]*w$, where $w < 1$.

Provided $w$, how many times must I use $f$ to sample from $X$ to decide whether $x_q \in X$ with some threshold confidence?

Note - If this problem is too open ended as things stand, please feel free to suggest additional restrictions or clarifications!

Note 2 - We can specify that $N \leq 100$, where $N$ is a positive integer, and that $w \leq \frac{1}{2}$.frac{1}{2}$, though we cannot say that$w << 1$. 6 added 76 characters in body; added 34 characters in body I have some set of known size but with unknown elements,$(x_1, ..., x_N) \in X$, where the elements of$X$are exponentially distributed random variables with unknown rate parameters,$(\lambda_1, ..., \lambda_N) \in R$. I also have a "black box" function$f$that samples an element from$X$with uniform probability, and then returns a randomly sampled value from the chosen element's exponential distribution (corresponding, perhaps, to the time until the first instance of an event governed by the chosen variable). I'm looking to use$f$to discern whether or not an exponentially distributed random variable,$x_q$, with known rate parameter,$\lambda_q$, exists in the set$X$. I also know that$\lambda_q$is smaller then all other rate parameters in the set$X$by at least a multiplicative factor$w$. Said another way,$\lambda_q \leq Min[(R-\lambda_q)]*w$, where$w < 1$. Provided$w$, how many times must I use$f$to sample from$X$to decide whether$x_q \in X$with some threshold confidence? Note - If this problem is too open ended as things stand, please feel free to suggest additional restrictions or clarifications! Note 2 - We can specify that$N \leq 100$, where$N$is a positive integer, and that$w \leq \frac{1}{2}$. 5 added 73 characters in body I have some set of known size but with unknown elements,$(x_1, ..., x_N) \in X$, where the elements of$X$are exponentially distributed random variables with unknown rate parameters,$(\lambda_1, ..., \lambda_N) \in R$. I also have a "black box" function$f$that samples an element from$X$with uniform probability, and then returns a randomly sampled value from the chosen element's exponential distribution (corresponding, perhaps, to the time until the first instance of an event governed by the chosen variable). I'm looking to use$f$to discern whether or not an exponentially distributed random variable,$x_q$, with known rate parameter,$\lambda_q$, exists in the set$X$. I also know that$\lambda_q$is smaller then all other rate parameters in the set$X$by at least a multiplicative factor$w$. Said another way,$\lambda_q \leq Min[(R-\lambda_q)]*w$, where$w < 1$. Provided$w$, how many times must I use$f$to sample from$X$to decide whether$x_q \in X\$ with some threshold confidence?

Note - If this problem is too open ended as things stand, please feel free to suggest additional restrictions or clarifications!

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