discrete random variable

(noun)

obtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, ….

Related Terms

  • probability mass function
  • random variable
  • probability distribution
  • expected value
  • standard deviation
  • continuous random variable
  • independent

Examples of discrete random variable in the following topics:

  • 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.
  • 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.
  • Two Types of Random Variables

    • A random variable $x$, and its distribution, can be discrete or continuous.
    • Random variables can be classified as either discrete (that is, taking any of a specified list of exact values) or as continuous (taking any numerical value in an interval or collection of intervals).
    • 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.
  • Probability Distribution Function (PDF) for a Discrete Random Variable

    • For a random sample of 50 mothers, the following information was obtained.
    • This is a discrete PDF because
  • Introduction

    • Continuous random variables have many applications.
    • The field of reliability depends on a variety of continuous random variables.
    • NOTE: The values of discrete and continuous random variables can be ambiguous.
    • For example, if X is equal to the number of miles (to the nearest mile) you drive to work, then X is a discrete random variable.
    • How the random variable is defined is very important.
  • Introduction

    • These two examples illustrate two different types of probability problems involving discrete random variables.
    • Recall that discrete data are data that you can count.
    • A random variable describes the outcomes of a statistical experiment in words.
    • The values of a random variable can vary with each repetition of an experiment.
    • In this chapter, you will study probability problems involving discrete random distributions.
  • 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.
    • The hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draws without replacement from a finite population of size $N$ containing a maximum of $K$ successes.
    • 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:
  • 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.
  • Recognizing and Using a Histogram

    • To define probability distributions for the simplest cases, one needs to distinguish between discrete and continuous random variables.
    • In the discrete case, one can easily assign a probability to each possible value.
    • In contrast, when a random variable takes values from a continuum, probabilities are nonzero only if they refer to finite intervals.
    • Intuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a discrete distribution, where the set of possible values for the random variable is, at most, countable.
    • If the distribution of $x$ is continuous, then $x$ is called a continuous random variable and, therefore, has a continuous probability distribution.
  • 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.
    • In the simple case of discrete time, a stochastic process amounts to a sequence of random variables known as a time series--for example, a Markov chain.
    • 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.
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