Mathematics
Probability Distribution
A probability distribution in mathematics is a function that describes the likelihood of different outcomes in a sample space. It assigns probabilities to each possible outcome, allowing for the calculation of expected values and the assessment of uncertainty in a given scenario. Probability distributions are fundamental in statistical analysis and are used to model random variables and events.
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eBook - PDF- Fetsje Bijma, Marianne Jonker, Aad van der Vaart, Reinie Erné(Authors)
- 2017(Publication Date)
- Amsterdam University Press(Publisher)
A Probability Theory A.1 Introduction This appendix contains some definitions and results from probability theory that are relevant when reading this book. The aim is to present this matter briefly. For further information, including proofs of theorems, examples, and applications, we refer to textbooks on probability theory, such as S. Ross, A First Course in Probability , Prentice-Hall and R. Meester, A Natural Introduction to Probability Theory , Birkh¨ auser. A.2 Distributions The basis of all statistical procedures is an observation X influenced by uncertainty, chance, or another form of randomness. As in probability theory, in statistics the uncertainty is translated mathematically by assigning a Probability Distribution to X . Definition A.1 Random variable A random variable is an observation subject to uncertainty, described by a Probability Distribution. 329 A: Probability Theory The set of all possible outcomes of X is called the sample space and denoted by Ω . A Probability Distribution is a function that gives the probability of finding the observation X in A for (almost) all subsets A ⊆ Ω , P( X ∈ A ) . Probability Distributions have three properties: (i) P( X ∈ Ω) = 1 . (ii) 0 ≤ P( X ∈ A ) ≤ 1 for all A ⊆ Ω . (iii) ( σ -additivity ) For disjoint sets A 1 , A 2 , . . . ⊆ Ω ( A i ∩ A j = ∅ if i � = j ), we have P( X ∈ ∞ uniondisplay i =1 A i ) = ∞ summationdisplay i =1 P( X ∈ A i ) . All other general properties of a Probability Distribution can be deduced directly from these three; for example, P( X ∈ A ) ≤ P( X ∈ B ) for A, B ⊆ Ω with A ⊆ B . For other properties of Probability Distributions, we refer to books on probability theory. In some cases, we are not interested in the random variable X itself, but in a function g of this variable, for example g ( X ) = X 2 . In many cases, the function g is defined on the entire real line, and g ( X ) is therefore defined for all real-valued random quantities X .
eBook - PDF- Jeffrey Camm, James Cochran, Michael Fry, Jeffrey Ohlmann(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Glossary 207 S U M M A R Y In this chapter we introduced the concept of probability as a means of understanding and measuring uncertainty. Uncertainty is a factor in virtually all business decisions, thus an understanding of probability is essential to modeling such decisions and improving the decision-making process. We introduced some basic relationships in probability including the concepts of out- comes, events, and calculations of related probabilities. We introduced the concept of conditional probability and discussed how to calculate posterior probabilities from prior probabilities using Bayes’ theorem. We then discussed both discrete and continuous ran- dom variables as well as some of the more common Probability Distributions related to these types of random variables. These Probability Distributions included the custom dis- crete, discrete uniform, binomial, and Poisson Probability Distributions for discrete random variables, as well as the uniform, triangular, normal, and exponential probability distri- butions for continuous random variables. We also discussed the concepts of the expected value (mean) and variance of a random variable. Probability is used in many chapters that follow in this textbook. The normal distribu- tion is essential to many of the predictive modeling techniques that we introduce in later chapters. Random variables and Probability Distributions will be seen again in Chapter 6 when we discuss the use of statistical inference to draw conclusions about a population from sample data, Chapter 7 when we discuss regression analysis as a way of estimating relationships between variables, and Chapter 11 when we discuss simulation as a means of modeling uncertainty. Conditional probability and Bayes’ theorem will be discussed in Chapter 15 in the context of decision analysis. It is very important to have a basic under- standing of probability, such as is provided in this chapter, as you continue to improve your skills in business analytics.
- Teresa Bradley(Author)
- 2014(Publication Date)
- Wiley(Publisher)
[ 206 ] C H A P T E R 5 5.1 Introduction to Probability Distributions and random variables The concept of a Probability Distribution was introduced briefly in Chapter 4 where it was described as a list of every possible outcome with the corresponding probability. The corresponding probabilities were calculated as simple relative frequencies. In Chapter 4 you may also have noted that probability calculations for even simple real life situations become complex very quickly! But, under certain well defined conditions it is possible to derive formulae (such as the Binomial formula introduced in 5.2) to calculate the probabilities. These formulae define ‘some of which are introduced in this chapter’. Some of the terminology associated with theoretical Probability Distributions is explained below. 5.1.1 Random variable In statistics the outcome of a random experiment is variable and determined by chance. The numeric value of the outcome is called a random variable. For example, when a die is thrown the face value that turns up may be any one of the numbers: 1, 2, 3, 4, 5 or 6. The face value that turns is a random variable. In other situations the outcome of a random experiment is not numeric, so a formula or rule is used to assign a numeric value to the each outcome. For example if three coins are tossed the outcomes are three Heads, two Heads and one Tail, one Head and two Tails or three Tails. In this experiment, a possible rule for assigning a numeric value to each outcome is to count the number of Heads. Then the random variable may have values 3, 2, 1 or 0: 3 (when the outcome is three Heads) 2 (when the outcome is two Heads and one Tail) 1 (when the outcome is one Head and two Tails) 0 (when the outcome is three Tails). Hence the more general definition of a random variable is the rule that assigns a numeric value to each outcome of a random experiment. The numeric values of a random variable may be discrete or continuous.
eBook - PDF- Hans-Otto Georgii(Author)
- 2012(Publication Date)
- De Gruyter(Publisher)
At this point, we need to emphasise that the term ‘distribution’ is used in an infla-tionary way in stochastics. Apart from the meaning that we have just introduced, it is also generally used as a synonym for probability measure. (In fact, every probability measure is the distribution of a random variable, namely the identity function of the underlying .) This has to be distinguished from two further notions, namely ‘distri-bution function’ and ‘distribution density’, which refer to the real event space . R ; B / and will be introduced now. Each probability measure P on . R ; B / is already uniquely determined by the func-tion F P .c/ WD P.Ł 1 ;cŁ/ for c 2 R . Likewise, the distribution of a real-valued random variable X on a probability space .; F ;P/ is uniquely determined by the function F X W c ! P.X c/ on R . This is because any two probability measures on . R ; B / coincide if and only if they agree on all intervals of the form Ł 1 ;cŁ , by statement (1.8d) and the uniqueness theorem (1.12). This motivates the following concepts. Section 1.3 Random Variables 23 Definition. For a probability measure P on the real line . R ; B / , the function F P W c ! P.Ł 1 ;cŁ/ from R to Œ0;1Ł is called the (cumulative) distribution function of P . Likewise, for a real random variable X on a probability space .; F ;P/ , the distribution function F X .c/ WD F P ı X 1 .c/ D P.X c/ of its distribution is called the (cumulative) distribution function of X . Every distribution function F D F X is increasing and right-continuous and has the asymptotic behaviour (1.29) lim c !1 F.c/ D 0 and lim c !C1 F.c/ D 1: This follows immediately from Theorem (1.11); see Problem 1.16. Figure 1.4 shows an example. Remarkably, every function with these properties is the distribution func-tion of a random variable on the unit interval (equipped with the uniform distribution from Example (1.19)).
- Will H. Moore, David A. Siegel(Authors)
- 2013(Publication Date)
- Princeton University Press(Publisher)
The probability that a random variable has any value is given by its proba- bility distribution; the support of this distribution is the set of all values for which the probability that the random variable takes on that value is greater than zero. This terminology is commonly used for continuous probability dis- tributions. Discrete random variables have discrete associated probability dis- 4 The connection between these two statements relates to the expectation of a random variable, which we discuss in Section 7 of this chapter. 5 Even theories that make deterministic claims typically highlight the way in which the claim varies with parameters of the model (i.e., independent variables). These comparative statics of theories, which we’ll discuss in Chapter 17 of this book, are weaker statements than the theory’s point predictions and more directly testable. Consequently, these are generally more believable. 6 A slightly more complex way of thinking about this is that a random variable stochas- tically determines that some event happens, out of all the events that might have happened, and assigns a value (typically a real number) to that event. 202 CHAPTER 10 Table 10.1: Top Problem Facing the United States, September 2001 Problem % Listing it No. 1 Terrorism 60 Economy 8 No opinion 7 Noneconomic, other 6 Moral decline 6 tributions; continuous random variables have continuous associated Probability Distributions. 10.2 SAMPLE DISTRIBUTIONS Sample distributions are empirical constructs. We distinguish them from Probability Distributions, which are constructed theoretically using the rules of classical probability. We discuss Probability Distributions below. 7 Sample distributions are representations of the number of cases that take each of the values in a sample space for a given portion of the population of cases.
eBook - PDF- Evert W. Johnson(Author)
- 2000(Publication Date)
- CRC Press(Publisher)
Rarely is the actual prob-ability distribution known. Consequently, persons carrying out such work must decide which known Probability Distribution most nearly matches what is known of the actual distribution. This decision is made using evidence such as empirical relative frequency distributions obtained from previous similar experiments or data obtained in the course of the current operation. When a Probability Distribution is used in this way, it is considered to be a model of the underlying true distribution. It may be a very good model, but it also may be very poor. In any case, it is being used as a representation of the true distribution because its characteristics are known and it is assumed that the unknown true distribution has similar characteristics. Needless to say, the matching of an appropriate Probability Distribution to a real-life situation, known as “choosing a model,” is of critical importance. 8.4 DISCRETE Probability DistributionS When the random variable can only assume a finite or countably infinite number of values, it is a discrete random variable and its Probability Distribution is referred to as a discrete Probability Distribution . The Probability Distributions associated with both the puppy and dice problems are of this type. In the case of discrete Probability Distributions, the probabilities associated with the individual values of the random variable are known as point probabilities and functions, such as Equation 8.4, when defined for 1 = 1, 2, … M , as point probability functions . When a discrete Probability Distribution is shown as a histogram , with the adjacent bars touching, the result is a figure similar to that in Figure 8.1, which shows the point probabilities associated with the possible outcomes when a pair of fair dice are tossed. Note that the bar widths ( D y ) are constant and that the bar heights ( N i ) indicate the number of outcomes that yield the indicated sum ( y i ).
eBook - ePubStatistics for Process Control Engineers
A Practical Approach
- Myke King(Author)
- 2017(Publication Date)
- Wiley(Publisher)
5 Probability Density FunctionThe probability density function (PDF) is a mathematical function that represents the distribution of a dataset. For example, if we were to throw a pair of unbiased six‐sided dice 36 times, we would expect (on average) the distribution of the total score to be that shown by Figure 5.1 . This shows the frequency distribution. To convert it to a Probability Distribution we divide each frequency by the total number of throws. For example, a total score of 5 would be expected to occur four times in 36 throws and so has a probability of 4/36 (about 0.111 or 11.1%). Figure 5.2 shows the resulting Probability Distribution.Expected frequency of total score from two diceFigure 5.1Expected distributionFigure 5.2Throwing dice generates a discrete distribution; in this case the result is restricted to integer values. Probability should not be plotted as continuous line. The probability of a non‐integer result is zero. But we can develop an equation for the line. In this case, if x is the total scored, the probability of scoring x is(5.1)Because x is discrete this function is known as the probability mass function (PMF). If the distribution were continuous we can convert the Probability Distribution to a probability density function (PDF) by dividing the probability by the range over which it applies.A condition of both functions is that the area they contain must be unity (or 100%) – in other words we are certain to be inside the area. So, in general(5.2)Provided this condition is met then any function can be described as a PDF (or PMF). We will show later that there are many functions that have a practical application. Unfortunately there are a larger number that appear to have been invented as a mathematical exercise and are yet to be shown that they describe any real probability behaviour.While the PMF allows us to estimate the probability of x having a certain value, the PDF does not. It only tells us the probability of x falling within a specified range. The probability of x being between a and b
- William DeCoursey(Author)
- 2003(Publication Date)
- Newnes(Publisher)
This chapter is concerned with discrete variables, and the next chapter is concerned with cases where the variable is continuous. Both types of variables are fundamental to some of the applications discussed in later chapters. In this chapter we will start with a general discussion of discrete random variables and their prob-ability and distribution functions. Then we will look at the idea of mathematical expectation, or the mean of a Probability Distribution, and the concept of the variance of a Probability Distribution. After that, we will look in detail at two important discrete Probability Distributions, the Binomial Distribution and the Poisson Distribution. 5.1 Probability Functions and Distribution Functions (a) Probability Functions Say the possible values of a discrete random variable, X , are x 0, x 1 , x 2 , ... x k , and the corresponding probabilities are p ( x 0 ), p ( x 1 ), p ( x 2 ) ... p ( x k ). Then for any choice of i , p ( x i ) ≥ 0, and ( ) 0 1 k i i p x = = ∑ , where k is the maximum possible value of i . Then p ( x i ) is a probability function , also called a probability mass function. An alternative nota-tion is that the probability function of X is written Pr [ X = x i ]. In many cases p ( x i ) (or Pr[ X = x i ]) and x i are related by an algebraic function, but in other cases the relation is shown in the form of a table. The relation can be represented by isolated spikes on a bar graph, as shown for example in Figure 5.1. By convention the random variable is represented by a capital letter (for example, X ), and particular values are represented by lower-case letters (for example, x , x i , x 0 ). Figure 5.1: Example of a Probability Function for a Discrete Random Variable 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Probability, p(x) -1 0 1 2 3 4 5 6 7 8 9 10 Value,x
- Aliakbar Montazer Haghighi, Indika Wickramasinghe(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
Note 3.6 Since the set consisting of all subsets of is extremely large, it will be impossible to assign probabilities to all. It has been shown in the theory of probability that a smaller set, say B, may be chosen that contains all events of our interest. In this case, B is referred to as the Borel field , similar to what was defined for discrete random variables. The triplet ( Ω , B, P ) is called the probability space , where P is the probability (measure) of events. 93 Random Variables Example 3.8 Suppose the temperatures in Houston in the month of July in the past many years have always been between 95 °F and 105 °F . This means that the temperature can take any value between the ranges 94.5 °F and 105.5 °F . When we say that the tem-perature is 100 °F , it means that the temperature lies somewhere between 99.5 °F and 100.5 °F . This is an example of a continuous random variable. 3.2 DISCRETE Probability Distribution (MASS) FUNCTIONS (PMF) A discrete random variable assigns discrete values to the sample points. As a ran-dom variable represents an event, we can assign probabilities to its values. A function describing how probabilities are distributed across the possible values of the random variable is referred to as the Probability Distribution of the random variable . In other words, in a random experiment, we might want to look at situations from which we can observe the predictable patterns and can use them as models. The patterns of such models are called the Probability Distributions . As a random variable may be discrete or continuous, so is its distribution. In this section, we discuss the discrete case. Note 3.7 A pmf of a random variable can be represented as a mathematical formula, table, or a graph. Definition 3.5 Let X be a discrete random variable (finite or countably infinite) defined on the sam-ple space Ω with range R x = { } 1 2 , x ,..., x n , where n is a nonnegative integer. Then, p X defined by p P X k = = { } X x , 1 k = ,2, ...
eBook - PDF- Haight(Author)
- 1963(Publication Date)
- Academic Press(Publisher)
In the remainder of this book, Probability Distributions will be theoretical, rather than empirical. Let us consider, therefore, some Probability Distributions which can be found by reasoning, without recourse to trial. If a fair die is thrown, the Probability Distribution is pl = p , mistakenly assumed that the values 1/6 could only be determined by experiment. Actually, the situation is quite the reverse. The list of six 1/6’s is in fact the definition of a fair die. The statistical problem then becomes one of deciding whether or not any particular die is fair. Any convergent series of positive constants can be used to form a Probability Distribution. We define three of these which are of importance in road traffic theory, and then show the types of traffic experiments which yield these distributions. = p -p - - - 9, = p , = 1/6. How is this known? It might be 6 PROBABILITY AND STATISTICS GEOMETRIC DISTRI B UTI ON. Consider the geometric series 1 + p + p 2 + p 3 + * * -. If p < 1, this is a convergent series, and has the sum 1/(1 - p). Therefore, a valid Probability Distribution can be defined simply by multiplying each term of the series by (1 - p), as follows: with the general expression p , = (1 - p)p, n = 0, 1, 2 , . . . . In Chapter 2, we shall see that an important quantity, the length of a queue, has this distribution. POISSON DISTRIBUTION. Consider the exponential series 1 + A + A 2 / 2 ! + A 3 / 3 ! + . . . , which is convergent for all A, and has the sum el. Dividing by eA, we obtain a valid Probability Distribution, defined as follows: with the general expression p , = A, e-lln!, n = 0, 1, 2,. . . . This distribution, as we shall prove later, describes the probability that exactly n randomly? arranged cars will be observed in unit length of road. t The word random is used in two ways in statistics: (a) as the equivalent In this book it will always of uncertain, and (b) a s the equivalent of Pozsson.
- Alan R. Jones(Author)
- 2018(Publication Date)
- Routledge(Publisher)
Not everything in life is certain, but sometimes it is important that we understand or at least have an appreciation of what might happen, so that we can put bounds around the extent of that uncertainty.That’s why estimators were created! We have already discussed that there are measures of Central Tendency, and of Dispersion and Shape to help us describe the degree of uncertainty and data clustering around some specific reference points. However, it is often helpful if we can articulate a more holistic picture of the range of potential values for a variable and just how likely or unlikely those values might occur. It would be better still, if we could relate them to some predefined patterns of Probability Distribution. In this chapter, we are going to discuss the properties of some of the more commonly used Probability Distributions in estimating; there are others – lots of them! As a conse- quence, this chapter cannot hope to cover all eventualities that might be pertinent to esti- mators and as such we should consider the discussions here to be a ‘Distribution Starter Kit’ – one that enables us to explore some of the useful key features of the more commonly used distributions and the circumstances in which it might be appropriate to use them. We can then make an informed choice of the appropriate distribution to use rather than rely on a single, unsophisticated ‘ one-size fits all’ default option that may have become custom and practice in some places but have little in the way of a true justification. ( By relying on the assumption of a single ‘one-size fits all’ distribution, be it Normal or Triangular, are we not risking becoming the unsophisticated forecaster in Lang’s analogy?) Probability Distributions 4 A word (or two) from the wise? ‘An unsophisticated forecaster uses statistics as a drunken man uses lamp-posts – for support rather than for illumination’. Attributed posthumously to: Andrew Lang 1840–1912 Author and poet
eBook - PDF- Ayanendranath Basu, Srabashi Basu(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
6 Random Variables and Probability Distributions So far we have described the probability of events which are elements of a sam- ple space in connection with a statistical experiment. In statistics, however, whenever we deal with probabilities, it is in relation to the possible values that a random variable can assume. So what is a random variable? To explain this concept, let us start with a simple probability experiment. Suppose we have a fair coin and we toss this coin three times in succession. Denoting the possible events of Head and Tail in each toss by H and T , the sample space of the entire experiment may be enumerated as S = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT }. As the coin is a fair one, each of these eight simple events has a probability of 1/8. Now, instead of finding the probabilities of each of the simple events in the sample space S separately, let us view the problem in a different way. Let us define a new variable X, which represents the number of heads in each of the simple events, and suppose our questions are now rephrased in terms of values of the variable X. For example, we may want to know the value of the probability Pr(X = 2). Since, according to our construction, each of the simple events {HHT }, {HTH} and {THH} corresponds to the event (X = 2), the required probability in this case is 18 + 18 + 18 = 38; similarly, Pr(X = 0) = 18, Pr(X = 1) = 38 and Pr(X = 3) = 18. Note that the variable X associates a real value with every element of the sample space. In the above example, the association induced by X can be represented as follows: HHH → 3 HHT → 2 HTH → 2 HTT → 1 THH → 2 THT → 1 TTH → 1 TTT → 0. This X is a random variable. Formally, a random variable is an association which matches every element in the sample space with a real value. In statis- tics, once we have defined a random variable, all our probability questions will be phrased in terms of the random variable, so much so that we will tend to 105
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