Mathematics
Discrete Random Variable
A discrete random variable is a variable that can take on a countable number of distinct values. These values are typically the result of counting or enumerating, such as the number of students in a class or the outcomes of rolling a die. The probability distribution of a discrete random variable can be described using a probability mass function.
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eBook - ePubIntroductory Probability and Statistics
Applications for Forestry and Natural Sciences (Revised Edition)
- Robert Kozak, Antal Kozak, Christina Staudhammer, Susan Watts(Authors)
- 2019(Publication Date)
- CAB International(Publisher)
discrete sample space is one that contains a finite number of elements, such as the eight possible outcomes from tossing a coin three times. A discrete sample space can also be unending, but countable, such as the sample space associated with tossing a coin until a head appears (the number of tosses necessary to meet this condition is the set of all possible whole numbers). Discrete Random Variables always take the form of data that are counted, such as the number of infested trees or the number of accidents per month in a logging camp.A continuous sample space is one that contains an infinite and uncountable number of outcomes. Any random variable obtained by measurements, like the time to germination, the weight of salmon, the distance between forest dependent com muni ties, or the volume of a tree, can theoretically take on any value in a measurement interval. For instance, for any two given merchantable tree volumes, e.g. 3.1 m3 and 3.2 m3 , one can always find another value that occurs between them (e.g. 3.17 m3 ). Theoretically, this could go on infinitely if measurement instruments were precise enough.Random variables defined over discrete sample spaces are called Discrete Random Variables , while random variables defined over continuous sample spaces are called continuous random variables .4.2 Probability Distributions
A Discrete Random Variable can be described by the probabilities that each of its individual values takes on when the random experiment is carried out. The list of all possible numerical outcomes and their associated probabilities is called the probability distribution of the random variable. For example, the probability distribution for the number of heads that occur when a coin is tossed three times is as follows (note that the random variable, number of heads, is denoted with an X , while the individual outcomes are denoted with an x
- George V Chilingar, Leonid F. Khilyuk, Herman H. Reike(Authors)
- 2012(Publication Date)
- Gulf Publishing Company(Publisher)
Fig. 7.1 .Figure 7.1 Typical graphs of distribution functions.Discrete Random VariableS
A set containing an infinite number of elements is called denumerable if all of its elements can be enumerated by natural numbers. For example, the set of all integers is denumerable because one can enumerate all the elements of the set in the following way:In this enumeration, one uses even numbers for positive integers and odd numbers for zero and negative integers. As an example of the set, which is not denumerable, one can mention the set of all real numbers from any interval [a , b ], where a < b .Definition 7.8. A set is called discrete if it contains a finite number of elements or is denumerable.For a complete description of any Discrete Random Variable, it is sufficient to give its possible values together with their probabilities (more accurately, the probabilities of events that consist in getting these values by random variables). Hence, a Discrete Random Variable can be defined by using a table such as Table 7.1 .Table 7.1 Distribution tableTables of such a kind are called distribution tables . They contain all possible values of a random variable in the upper row and corresponding probabilities in the lower row; more precisely the probabilities of events (ξ =xk), k = 1, 2, …, n. It is convenient to introduce the function p (x )asThe function p (x
- Mikel J. Harry, Prem S. Mann, Ofelia C. De Hodgins, Richard L. Hulbert, Christopher J. Lacke(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
Many other variables are continuous, especially those that involve some measure of weight, volume, distance, or area. Often continuous random variables are used to approximate distributions that involve money, even though it is a Discrete Random Variable. This occurs when there is a large set of possible values, such as the price of a house. Following are a few more examples of continuous random variables:1. The weight of an outgoing shipment 2. The amount of gas dispensed when the fuel pump states one gallon 3. The distance traveled by a delivery truck in a single day 4. The price of a gallon of gas This chapter is limited to discussion of Discrete Random Variables. Chapter 12 contains discussion of continuous random variables. 11.5 PROBABILITY DISTRIBUTIONS OF A Discrete Random VariableLet x be a Discrete Random Variable. The probability distribution of x describes how the probability is distributed over the possible values of x . Example 11.3 demonstrates the concept of a probability distribution of a Discrete Random Variable by extending Example 11.2.Example 11.3 Recall from Example 11.2 the outcomes and values of x corresponding to the number of customers out of four who make a deposit. This information is reproduced in Table 11.3 . In Example 11.2, we did not make any assumptions regarding how likely a customer is to make a deposit, but we will provide an example here. In addition, we will make a common assumption that customers arriving at a bank make deposits independently of each other.Suppose that there is a probability of .50 that any given customer makes a deposit. On the basis of this information, Table 11.3 lists the various values of x, the corresponding outcomes, and the probabilities of various values of x . Note that this example contains 16 outcomes and all of them are equally likely (because of .50 probability of deposit), so that the probability of any specific value of x
eBook - ePub- Ayanendranath Basu, Srabashi Basu(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
6.1 Discrete and Continuous Random VariablesA random variable, therefore, is characterized by the set of possible values it can assume together with the probability structure on this set. If the random variable can only assume finitely many distinct values, or at most countably many distinct values x1 , x2 ,…, the random variable is called a Discrete Random Variable. Clearly, such a random variable concentrates the entire probability on a set of discrete points. The set of possible values that the random variable can assume is called its support, and this is denoted by the symbol χ in the previous subsection. The random variables defined in Examples 6.1 and 6.2 are both Discrete Random Variables.x 1,x 2, … ,x kFor a Discrete Random Variable X supported on χ, we define a function f, referred to as the probability mass function (PMF) of X, asf ( x ) = {P r ( X = x )for x ∈ Χ ,0otherwise .Often we simply define the PMF only on the support, it being implicitly understood that it is equal to zero everywhere else. This function f together with the support χ completely characterizes the random variable X. Note that the PMF f(x) satisfies the following two conditions:( i ) f ( x ) ≥ 0 for all x , and ( i i )∑f ( x ) = 1.x ∈ χExample 6.3. The motivating example with which we began this chapter included tossing a fair coin three times and counting the number of heads (which is our random variable X). In this case the support of the random variable X
eBook - ePubProbability, Statistics, and Data
A Fresh Approach Using R
- Darrin Speegle, Bryan Clair(Authors)
- 2021(Publication Date)
- Chapman and Hall/CRC(Publisher)
} and so:P ( X = 2 ) = P ( { H H T , H T H , T H H } ) =3 8.It is often easier, both notationally and for doing computations, to hide the sample space and focus only on the random variable. We will not always explicitly define the sample space of an experiment. It is easier, more intuitive and (for the purposes of this book) equivalent to just understandP ( a < X < b )for all choices ofa < b. By understanding these probabilities, we can derive many useful properties of the random variable, and hence, the underlying experiment.We will consider two types of random variables in this book. Discrete Random Variables are integers, and often come from counting something. Continuous random variables take values in an interval of real numbers, and often come from measuring something. Working with Discrete Random Variables requires summation, while continuous random variables require integration. We will discuss these two types of random variables separately in this chapter and in Chapter 4 .3.1 Probability mass functions
Definition 3.2. A discrete random variable is a random variable that takes integer values.1 A Discrete Random Variable is characterized by its probability mass function (pmf).The pmf p of a random variable X is given byp ( x ) = P ( X = x ) .The pmf may be given in table form or as an equation. Knowing the probability mass function determines the Discrete Random Variable, and we will understand the random variable by understanding its pmf.Probability mass functions satisfy the following properties:To check that a function is a pmf, we check that all of its values are probabilities, and that those values sum to one.Theorem 3.1. Let p be the probability mass function of X.- p ( x ) ≥ 0for all x.
- .( x ) = 1∑ xp
Example 3.2. The Eurasian lynx is a wild cat that lives in the north of Europe and Asia. When a female lynx gives birth, she may have from 1 to 4 kittens. Our statistical experiment is a lynx giving birth, and the outcome is a litter of baby lynx. Baby lynx are complicated objects, but there is a simple random variable here: the number of kittens. Call this X. Ecologists have estimated2 the pmf for X
eBook - ePubAn Introduction to Financial Mathematics
Option Valuation
- Hugo D. Junghenn(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 3 Random Variables 3.1 IntroductionWe have seen that outcomes of experiments are frequently real numbers. Such outcomes are called random variables. Before giving a formal definition, we introduce a convenient shorthand notation for describing sets involving real-valued functions X, Y etc. on Ω. The notation essentially leaves out the standard “ω ∈ Ω” part of the description, which is often redundant. Some typical examples are{≔X < a}{andω ∈ Ω | X}( ω )< a{≔X ≤ Y}{.ω ∈ Ω | X}( ω )≤ Y( ω )We also use notation such as {X ∈ A, Y ∈ B} for {X ∈ A} ∩ {Y ∈ B}. For probabilities we write, for example,ℙrather than(X < a)ℙ. The following numerical example should illustrate the basic idea.({)X < a}3.1.1 Example. The table below summarizes the distribution of grade-point averages for a group of 100 students. The first row gives the number of students having the grade-point averages listed in the second row. If X denotes the grade-point average of a randomly chosen student, then, using the above notation, we see that the probability that a student has a grade point average greater than 2.5 but not greater than 3.3 isℙ(2.5 < X ≤ 3.3)= ℙ(+ ℙX = 2.7)(+ ℙX = 2.9)(+ ℙX = 3.1)(X = 3.3)= .16 + .12 + .10 + .08 = .46 .The preceding example suggests that is useful to be able to assign probabilities to sets of the form {X ∈ J}, where X is a numerical outcome of an experiment and J is an interval. Accordingly, we define a random variable on a probability space(to be a function X : Ω → R such that for each interval J the set {X ∈ J} is a member of ℱ . In case of ambiguity, we refer to X as an ℱ -random variable or say that X is ℱ -measurable.Ω , ℱ , ℙ)An important example of a random variable, one that appears throughout the text, is the price of a stock at a prescribed time. The underlying probability space in this case must be carefully constructed and can take several forms. Particular concrete models are described in later chapters.
eBook - ePub- Mustapha Abiodun Akinkunmi(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
5 Discrete Probability Distributions A random variable (X) that takes integer values, X 1, X 2, …, X n with the corresponding probabilities of P (X 1), P (X 1), …, P (X n) and the probabilities P X such that ∑ 1 n P X = 1 is called a discrete probability distribution. The type of the random variable determines the nature of the probability distribution it follows. A Discrete Random Variable is usually an integer value while a continuous random variable involves measuring and takes both integer values and a fractional part or real number. When the probabilities are assigned to random variables, then the collection of such probabilities give rise to a probability distribution. The probability distribution function can be abbreviated as pdf. In this book, we shall be using both the abbreviation and the full words. A discrete probability distribution satisfies two conditions: (5.1) 0 ≤ P X ≤ 1 a n d (5.2) ∑ P X = 1 5.1 Probability Mass Distribution The probability that a Discrete Random Variable X takes on a particular value x (i.e., P X = x = f x) is called a probability mass function (pmf) if it satisfies the following: P X = x = f x > 0, i f x ∈ S All probability must be positive for every element x in the sample space S. Hence, if element x is not in the sample space S, then f x = 0. ∑ x ∈ S f x = 1 The sum of probabilities for all of the possible x values in the sample space S must equal 1. P X ∈ x = ∑ x ∈ A f x The sum of probabilities of the x values in A is the probability of event A. Example 5.1: Experiment: Toss a fair coin 2 times Sample space: S = H H, H T, T H, T T Random variable X is the number of tosses showing. heads Thus X : S → R X = H H = 2 X = H T = T H = 1 X = T T = 0 X = 0, 1, 2. That is, random variable X takes a range of values 0, 1, and 2. Hence, the probability mass function is given by: P X = 0 = 1 4, P X = 1 = 1 2, a n d P X = 2 = 1 4 Example 5.2: Suppose a real estate agent sold a number of houses in a month
eBook - ePub- Maria L. Rizzo(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 2 Probability and Statistics ReviewIn this chapter we briefly review without proofs some definitions and concepts in probability and statistics. Many introductory and more advanced texts can be recommended for review and reference. On introductory probability see e.g., Bean [26 ], Ghahramani [124 ], or Ross [249 ]. Mathematical statistics and probability books at an advanced undergraduate or first-year graduate level include, e.g., DeGroot and Schervish [69 ], Freund (Miller and Miller) [209 ], Hogg, McKean and Craig [150 ] or Larsen and Marx [177 ]. Casella and Berger [40 ] or Bain and Englehart [21 ] are somewhat more advanced. Durrett [80 ] is a graduate probability text. Lehmann [180 ] and Lehmann and Casella [181 ] are graduate texts in statistical inference.2.1 Random Variables and Probability Distribution and Density FunctionsThe cumulative distribution function (cdf) of a random variable X is FX defined byF X( x ) = P ( X ≤ x ) , x ∈ ℝ .In this book P(·) denotes the probability of its argument. We will omit the subscript X and write F(x) if it is clear in context. The cdf has the following properties:1. FX is non-decreasing.2. FX is right-continuous; that is,limϵ →0 +F X( x + ϵ ) =F X( x ) , for all x ∈ ℝ .3.andlimx → − ∞F X( x ) = 0.limx → ∞F X( x ) = 1A random variable X is continuous if FX is a continuous function. A random variable X is discrete if FX is a step function.Discrete distributions can be specified by the probability mass function (pmf) pX (x) = P(X = x). The discontinuities in the cdf are at the points where the pmf is positive, and p(x) = F(x) − F(x− ).If X is discrete, the cdf of X isF X( x ) = P ( X ≤ x ) =( k ) .∑{ k ≤ x :p X( k ) > 0 }p XContinuous distributions do not have positive probability mass at any single point. For continuous random variables X the probability density function (pdf) or density of X is, provided that FXf X( x )=F X '( x )
