random variable

Examples of random variable in the following topics:

• 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.

• Two Types of Random Variables

  • A random variable $x$, and its distribution, can be discrete or continuous.
  • As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability.
  • 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.

• Expected Values of Discrete Random Variables

  • The expected value of a random variable is the weighted average of all possible values that this random variable can take on.
  • A discrete random variable $X$ has a countable number of possible values.
  • The probability distribution of a discrete random variable $X$ lists the values and their probabilities, such that $x_i$ has a probability of $p_i$.
  • In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or first moment) of a random variable is the weighted average of all possible values that this random variable can take on.
  • The weights used in computing this average are probabilities in the case of a discrete random variable.

• 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.

• Probability Distributions for Discrete Random Variables

  • Probability distributions for discrete random variables can be displayed as a formula, in a table, or in a graph.
  • A discrete random variable $x$ has a countable number of possible values.
  • The probability mass function has the same purpose as the probability histogram, and displays specific probabilities for each discrete random variable.
  • This histogram displays the probabilities of each of the three discrete random variables.
  • This table shows the values of the discrete random variable can take on and their corresponding probabilities.

• The Central Limit Theorem for Sums

  • Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:
  • If you draw random samples of size n, then as n increases, the random variable ΣX which consists of sums tends to be normally distributed and

• The Hypergeometric Random Variable

  • A hypergeometric random variable is a discrete random variable characterized by a fixed number of trials with differing probabilities of success.
  • As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw.
  • A random variable follows the hypergeometric distribution if its probability mass function is given by:

• Bernoulli distribution

  • A Bernoulli random variable has exactly two possible outcomes.
  • Bernoulli random variables are often denoted as 1 for a success and 0 for a failure.
  • This mathematical inquiry of Bernoulli random variables can be extended even further.
  • If X is a random variable that takes value 1 with probability of success p and 0 with probability 1−p, then X is a Bernoulli random variable with mean and standard deviation µ = p, σ =$\sqrt{p(1-p)}$
  • In general, it is useful to think about a Bernoulli random variable as a random process with only two outcomes: a success or failure.

• Chance Processes

  • A stochastic process is a collection of random variables that is often used to represent the evolution of some random value over time.
  • In probability theory, a stochastic process--sometimes called a random process-- is a collection of random variables that is often used to represent the evolution of some random value, or system, over time.
  • Random variables are non-deterministic (single) quantities which have certain probability distributions.
  • Random variables corresponding to various times (or points, in the case of random fields) may be completely different.
  • Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical correlations.

• Randomized Design: Single-Factor

  • Completely randomized designs study the effects of one primary factor without the need to take other nuisance variables into account.
  • In the design of experiments, completely randomized designs are for studying the effects of one primary factor without the need to take into account other nuisance variables.
  • The experiment under a completely randomized design compares the values of a response variable based on the different levels of that primary factor.
  • An example of a completely randomized design using the three numbers is:
  • Discover how randomized experimental design allows researchers to study the effects of a single factor without taking into account other nuisance variables.