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Random Variables01:09

Random Variables

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A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase 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.
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Sampling is a technique to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population. The sampling method ensures that samples are drawn without bias and accurately represent the population. Because measuring the entire population in a study is not practical, researchers use samples to represent the population of interest. Among the various sampling methods used by...
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The randomization process involves assigning study participants randomly to experimental or control groups based on their probability of being equally assigned. Randomization is meant to eliminate selection bias and balance known and unknown confounding factors so that the control group is similar to the treatment group as much as possible. A computer program and a random number generator can be used to assign participants to groups in a way that minimizes bias.
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Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant.
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Cluster Sampling Method01:20

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Appropriate sampling methods ensure that samples are drawn without bias and accurately represent the population. Because measuring the entire population in a study is not practical, researchers use samples to represent the population of interest.
To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four departments from your...
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Poisson Probability Distribution01:09

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A Poisson probability distribution is a discrete probability distribution. It gives the probability of a number of events occurring in a fixed interval of time or space if these events happen at a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on average, there are five words spelled incorrectly in 100 pages. The interval is 100 pages.
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Related Experiment Video

Updated: Apr 4, 2026

Modeling the Functional Network for Spatial Navigation in the Human Brain
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Fast Generation of Sparse Random Kernel Graphs.

Aric Hagberg1, Nathan Lemons2

  • 1Center for Nonlinear Studies, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America.

Plos One
|September 11, 2015
PubMed
Summary
This summary is machine-generated.

We introduce random kernel graphs, a flexible model for complex networks. Our efficient algorithm generates large sparse graphs with tunable properties in near-linear time.

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Area of Science:

  • Network Science
  • Graph Theory
  • Computational Mathematics

Background:

  • Kernel-based inhomogeneous random graphs offer flexibility and mathematical tractability for modeling real-world networks.
  • Existing models often struggle with scalability for large networks.

Purpose of the Study:

  • To introduce a novel class of inhomogeneous random graph models called random kernel graphs.
  • To develop an efficient algorithm for generating sparse random kernel graphs with tunable properties.
  • To demonstrate the algorithm's scalability for large networks.

Main Methods:

  • Specification of the random kernel graph model.
  • Development of an efficient graph generation algorithm.
  • Analysis of algorithm runtime complexity.
  • Application to generating power-law degree distribution graphs with tunable assortativity.

Main Results:

  • The proposed random kernel graph model produces sparse graphs with controllable properties.
  • The generation algorithm achieves a runtime complexity of at most O(n(logn)^2) for many practical kernels.
  • The algorithm efficiently generates samples of power-law degree distribution graphs with tunable assortativity.

Conclusions:

  • Random kernel graphs provide a powerful and efficient framework for modeling complex networks.
  • The developed algorithm offers a scalable solution for generating large, realistic network instances.
  • This approach facilitates the study of network properties like degree distribution and assortativity.