random walk

(noun)

a stochastic path consisting of a series of sequential movements, the direction (and sometime length) of which is chosen at random

Related Terms

Examples of random walk in the following topics:

  • 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.

  • 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.

  • Two Types of Random Variables

    • A random variable $x$, and its distribution, can be discrete or continuous.
    • They may also conceptually represent either the results of an "objectively" random process (such as rolling a die), or the "subjective" randomness that results from incomplete knowledge of a quantity.
    • Discrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers).
    • Examples of discrete random variables include the values obtained from rolling a die and the grades received on a test out of 100.
    • Selecting random numbers between 0 and 1 are examples of continuous random variables because there are an infinite number of possibilities.

  • Randomization for two-way tables and chi-square

    • Randomization methods may also be used for the contingency tables.
    • In short, we create a randomized contingency table, then compute a chi-square test statistic.
    • This randomization approach is valid for any sized sample, and it will be more accurate for cases where one or more expected bin counts do not meet the minimum threshold of 5.

  • Randomized Design: Single-Factor

    • Completely randomized designs study the effects of one primary factor without the need to take other nuisance variables into account.
    • For completely randomized designs, the levels of the primary factor are randomly assigned to the experimental units.
    • In complete random design, the run sequence of the experimental units is determined randomly.
    • To randomize the runs, one way would be to put 6 slips of paper in a box with 2 having level 1, 2 having level 2, and 2 having level 3.
    • An example of a completely randomized design using the three numbers is:

  • Probability

    • We use probability to build tools to describe and understand apparent randomness.
    • Random processes include rolling a die and flipping a coin. ( a) Think of another random process.
    • What we think of as random processes are not necessarily random, but they may just be too difficult to understand exactly.
    • However, even if a roommate's behavior is not truly random, modeling her behavior as a random process can still be useful.
    • It can be helpful to model a process as random even if it is not truly random.

  • Random Variable Notation

    • Upper case letters like X or Y denote a random variable.
    • Lower case letters like x or y denote the value of a random variable.
    • If X is a random variable, then X is written in words. and x is given as a number.
    • Because you can count the possible values that X can take on and the outcomes are random (the x values 0, 1, 2, 3), X is a discrete random variable.

  • Introduction

    • Continuous random variables have many applications.
    • The field of reliability depends on a variety of continuous random variables.
    • This chapter gives an introduction to continuous random variables and the many continuous distributions.
    • NOTE: The values of discrete and continuous random variables can be ambiguous.
    • How the random variable is defined is very important.