discrete random variable

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.