2002/09/30 by R. K. P. Zia, B. Schmittmann · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Applied mathematics #Complex Systems and Time Series Analysis #Connection (principal bundle) #Context (archaeology) #Distribution (mathematics) #Geometry #Mathematical analysis #Mathematics #Observable #Perspective (graphical) #Physics #Probability and statistics #Probability distribution #Quantum mechanics #Simple (philosophy) #Standard deviation #Statistical Mechanics and Entropy #Statistical physics #Statistics #Stochastic processes and statistical mechanics #String (physics) #Theoretical physics #Variance (accounting) #cond-mat
paper · pdf · doi:10.1119/1.1566430
8 pages, 2 figures. Some typos corrected in discussion of Fig. 1. Pedagogical article, written for the American Journal of Physics
arxiv created 2003/03/24 · openalex publication_date 2003/08/12 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
For any physical observable in statistical systems, the most frequently studied quantities are its average and standard deviation. Yet, its full distribution often carries extremely interesting information and can be invoked to put the properties of the individual moments into perspective. As an example, we consider a problem concerning simple random walks. When a drunk is observed over L nights, taking N steps per night, and the number of steps to the right is recorded for each night, an average and a variance based on these data can be calculated. When the variance is used to estimate p, the probability for the drunk to step right, complex values for p are frequently found. To put such obviously nonsensical results into context, we study the full probability distribution for the variance of the data string. We discuss the connection of our results to the problem of data binning and provide two other examples to demonstrate the importance of full distributions.