random walk

Examples of random walk in the following topics:

• A Random Walk

  • Currency market exchange rates could exhibit a random walk in the short run.
  • Statisticians define a random walk whose current value is the previous period's value plus a random disturbance.
  • We show a random walk in Equation 1.
  • A random walk has an unique characteristic – the variable drifts in a particular direction before changing direction.
  • Many exchange rates exhibit this structure or exhibit a pure random walk. ) However, these statistical tests are beyond the scope of this book.

• Chance Processes

  • In the case of a random walk, for example, the law is the probability distribution of the possible trajectories of the walk.
  • A random walk is a mathematical formalization of a path that consists of a succession of random steps.
  • Thus, the random walk serves as a fundamental model for recorded stochastic activity.
  • Example of eight random walks in one dimension starting at 0.
  • Summarize the stochastic process and state its relationship to random walks.

• Behavior of an Efficient Market

  • Historically, there was a very close link between EMH and the random-walk model and then the Martingale model.
  • The random character of stock market prices was first modelled by Jules Regnault, a French broker, in 1863 and then by Louis Bachelier, a French mathematician, in his 1900 PhD thesis, "The Theory of Speculation. " His work was largely ignored until the 1950's; however, beginning in the 1930's scattered, independent work corroborated his thesis.
  • A small number of studies indicated that U.S. stock prices and related financial series followed a random walk model.
  • In 1965 Eugene Fama published his dissertation arguing for the random walk hypothesis, and Samuelson published a proof for a version of the efficient-market hypothesis.

• Random Walks

  • We can get an order of magnitude feeling for how much scattering will affect the radiation field emerging from a source using the concepts if the mean free path and the random walk.

• Chemotaxis

  • If one watches a bacterium swimming in a uniform environment, its movement will look like a random walk with relatively straight swims interrupted by random tumbles that reorient it.
  • If it is moving in the wrong direction, it will tumble sooner and try a new direction at random.
  • This is a result of simply choosing between two methods of random movement; namely tumbling and straight swimming.

• Comparing Two Populations: Independent Samples

  • Nonparametric methods for testing the independence of samples include Spearman's rank correlation coefficient, the Kendall tau rank correlation coefficient, the Kruskal–Wallis one-way analysis of variance, and the Walk–Wolfowitz runs test.
  • Let $(x_1, y_1), (x_2, y_2), \cdots, (x_n, y_n)$ be a set of observations of the joint random variables $X$ and $Y$ respectively, such that all the values of ($x_i$) and ($y_i$) are unique.
  • The Walk–Wolfowitz runs test is a non-parametric statistical test that checks a randomness hypothesis for a two-valued data sequence.
  • Contrast Spearman, Kendall, Kruskal–Wallis, and Walk–Wolfowitz methods for examining the independence of samples

• Variability in linear combinations of random variables

  • Quantifying the average outcome from a linear combination of random variables is helpful, but it is also important to have some sense of the uncertainty associated with the total outcome of that combination of random variables.
  • The variance of a linear combination of random variables can be computed by plugging in the variances of the individual random variables and squaring the coefficients of the random variables:
  • The variance of a linear combination of random variables may be computed by squaring the constants, substituting in the variances for the random variables, and computing the result:
  • This equation is valid as long as the random variables are independent of each other.
  • However, if John walks to work, then his commute is probably not affected by any weekly traffic cycle.

• Sampling from a population

  • The most basic type of random selection is equivalent to how raffles are conducted.
  • The selected names would represent a random sample of 100 graduates.
  • The most basic random sample is called a simple random sample, and which is equivalent to using a raffle to select cases.
  • The act of taking a simple random sample helps minimize bias, however, bias can crop up in other ways.
  • For instance, if a political survey is done by stopping people walking in the Bronx, this will not represent all of New York City.

• Basic Inferential Statistics

  • Such populations include, for example, the length of all supermarket checkout lines, the height of every person in the world, or the age at which every child on Earth first began to walk.
  • Researchers will ideally want to strive to ensure that their sample is truly random.
  • In a “random sample,” every element of the population has an equal chance of being included in the sample.
  • Although random samples are the ideal, researchers will often end up using samples of convenience instead (e.g., volunteers from an Introduction to Psychology class) because truly random samples are difficult to obtain and often impractical.
  • We would first collect our sample from the population, ideally a random one.

• Walks etc.

  • A closed walk is one where the beginning and end point of the walk are the same actor.
  • Walks are unrestricted.
  • There is one walk of length 2 (A,B,C).
  • All trails are walks, but not all walks are trails.
  • All paths are trails and walks, but all walks and all trails are not paths.