|
7
|
|
|
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
|
|
|
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
|
|
|
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!
|
|
|
|
4
|
|
|
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 $w$.
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!
|
|
|
|
|
3
|
|
|
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 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 $w$.
Provided $w$, how many times must I use $f$ to sample from $X$ to decide whether $x_q \in X$ with some threshold confidence?
|
|
|
|
|
2
|
|
|
I have some set of unknown 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 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 $w$.
Provided $w$, how many times must I use $f$ to sample from $X$ to decide whether $x_q \in X$ with some threshold confidence?
|
|
|
|
|
1
|
|
|
Deciding whether or not an exponentially distributed random variable exists in a set via the use of a "black box" function
I have some set of unknown size, $(x_1, ..., x_N) \in X$, 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 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 $w$.
Provided $w$, how many times must I use $f$ to sample from $X$ to decide whether $x_q \in X$ with some threshold confidence?
|
|
|
