@article{eryilmaz_2010, 
  title={Discrete Scan Statistics Generated by Exchangeable Binary Trials}, 
  volume={47}, 
  DOI={10.1017/s0021900200007385}, 
  abstractNote={<jats:p>Let {<jats:italic>X</jats:italic>
               <jats:sub>
                  <jats:italic>i</jats:italic>
               </jats:sub>}<jats:sub>
                  <jats:italic>i</jats:italic>=1</jats:sub>
               <jats:sup>
                  <jats:italic>n</jats:italic>
               </jats:sup> be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable <jats:italic>S</jats:italic>
               <jats:sub>
                  <jats:italic>n</jats:italic>,<jats:italic>m</jats:italic>
               </jats:sub> to be the maximum number of 1s within any <jats:italic>m</jats:italic> consecutive trials in {<jats:italic>X</jats:italic>
               <jats:sub>
                  <jats:italic>i</jats:italic>
               </jats:sub>}<jats:sub>
                  <jats:italic>i</jats:italic>=1</jats:sub>
               <jats:sup>
                  <jats:italic>n</jats:italic>
               </jats:sup>. The random variable <jats:italic>S</jats:italic>
               <jats:sub>
                  <jats:italic>n</jats:italic>,<jats:italic>m</jats:italic>
               </jats:sub> is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {<jats:italic>X</jats:italic>
               <jats:sub>
                  <jats:italic>i</jats:italic>
               </jats:sub>}<jats:sub>
                  <jats:italic>i</jats:italic>=1</jats:sub>
               <jats:sup>
                  <jats:italic>n</jats:italic>
               </jats:sup> consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of <jats:italic>S</jats:italic>
               <jats:sub>
                  <jats:italic>n</jats:italic>,<jats:italic>m</jats:italic>
               </jats:sub> for 2<jats:italic>m</jats:italic> ≥ <jats:italic>n</jats:italic>. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.</jats:p>}, 
  number={04}, 
  publisher={Cambridge University Press (CUP)}, 
  author={Eryilmaz}, 
  year={2010}
  }