Is the Cosmos Random?
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Random Variable = a Real-Valued Function of an Outcome X = F(Outcome)
Random Variables (Chapter 2) Random variable = A real-valued function of an outcome X = f(outcome) Domain of X: Sample space of the experiment. Ex: Consider an experiment consisting of 3 Bernoulli trials. Bernoulli trial = Only two possible outcomes – success (S) or failure (F). • “IF” statement: if … then “S” else “F” • Examine each component. S = “acceptable”, F = “defective”. • Transmit binary digits through a communication channel. S = “digit received correctly”, F = “digit received incorrectly”. Suppose the trials are independent and each trial has a probability ½ of success. X = # successes observed in the experiment. Possible values: Outcome Value of X (SSS) (SSF) (SFS) … … (FFF) Random variable: • Assigns a real number to each outcome in S. • Denoted by X, Y, Z, etc., and its values by x, y, z, etc. • Its value depends on chance. • Its value becomes available once the experiment is completed and the outcome is known. • Probabilities of its values are determined by the probabilities of the outcomes in the sample space. Probability distribution of X = A table, formula or a graph that summarizes how the total probability of one is distributed over all the possible values of X. In the Bernoulli trials example, what is the distribution of X? 1 Two types of random variables: Discrete rv = Takes finite or countable number of values • Number of jobs in a queue • Number of errors • Number of successes, etc. Continuous rv = Takes all values in an interval – i.e., it has uncountable number of values. • Execution time • Waiting time • Miles per gallon • Distance traveled, etc. Discrete random variables X = A discrete rv. -
Indeterminism in Physics and Intuitionistic Mathematics
Indeterminism in Physics and Intuitionistic Mathematics Nicolas Gisin Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland (Dated: November 5, 2020) Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to “speak” of indeterminism, its inability to present us a worldview in which new information is created as time passes. In such a case, scientific determinism would only be an illusion due to the timeless mathematical language scientists use. To investigate this possibility it is necessary to develop an alternative mathematical language that is both powerful enough to allow scientists to compute predictions and compatible with indeterminism and the passage of time. We argue that intuitionistic mathematics provides such a language and we illustrate it in simple terms. I. INTRODUCTION I have always been amazed by the huge difficulties that my fellow physicists seem to encounter when con- templating indeterminism and the highly sophisticated Physicists are not used to thinking of the world as in- circumlocutions they are ready to swallow to avoid the determinate and its evolution as indeterministic. New- straightforward conclusion that physics does not neces- ton’s equations, like Maxwell’s and Schr¨odinger’s equa- sarily present us a deterministic worldview. But even tions, are (partial) differential equations describing the more surprising to me is the attitude of most philoso- continuous evolution of the initial condition as a func- phers, especially philosophers of science. Indeed, most tion of a parameter identified with time. -
Wittgenstein on Freedom of the Will: Not Determinism, Yet Not Indeterminism
Wittgenstein on Freedom of the Will: Not Determinism, Yet Not Indeterminism Thomas Nadelhoffer This is a prepublication draft. This version is being revised for resubmission to a journal. Abstract Since the publication of Wittgenstein’s Lectures on Freedom of the Will, his remarks about free will and determinism have received very little attention. Insofar as these lectures give us an opportunity to see him at work on a traditional—and seemingly intractable—philosophical problem and given the voluminous secondary literature written about nearly every other facet of Wittgenstein’s life and philosophy, this neglect is both surprising and unfortunate. Perhaps these lectures have not attracted much attention because they are available to us only in the form of a single student’s notes (Yorick Smythies). Or perhaps it is because, as one Wittgenstein scholar put it, the lectures represent only “cursory reflections” that “are themselves uncompelling." (Glock 1996: 390) Either way, my goal is to show that Wittgenstein’s views about freedom of the will merit closer attention. All of these arguments might look as if I wanted to argue for the freedom of the will or against it. But I don't want to. --Ludwig Wittgenstein, Lectures on Freedom of the Will Since the publication of Wittgenstein’s Lectures on Freedom of the Will,1 his remarks from these lectures about free will and determinism have received very little attention.2 Insofar as these lectures give us an opportunity to see him at work on a traditional—and seemingly intractable— philosophical problem and given the voluminous secondary literature written about nearly every 1 Wittgenstein’s “Lectures on Freedom of the Will” will be abbreviated as LFW 1993 in this paper (see bibliography) since I am using the version reprinted in Philosophical Occasions (1993). -
Topic 1: Basic Probability Definition of Sets
Topic 1: Basic probability ² Review of sets ² Sample space and probability measure ² Probability axioms ² Basic probability laws ² Conditional probability ² Bayes' rules ² Independence ² Counting ES150 { Harvard SEAS 1 De¯nition of Sets ² A set S is a collection of objects, which are the elements of the set. { The number of elements in a set S can be ¯nite S = fx1; x2; : : : ; xng or in¯nite but countable S = fx1; x2; : : :g or uncountably in¯nite. { S can also contain elements with a certain property S = fx j x satis¯es P g ² S is a subset of T if every element of S also belongs to T S ½ T or T S If S ½ T and T ½ S then S = T . ² The universal set is the set of all objects within a context. We then consider all sets S ½ . ES150 { Harvard SEAS 2 Set Operations and Properties ² Set operations { Complement Ac: set of all elements not in A { Union A \ B: set of all elements in A or B or both { Intersection A [ B: set of all elements common in both A and B { Di®erence A ¡ B: set containing all elements in A but not in B. ² Properties of set operations { Commutative: A \ B = B \ A and A [ B = B [ A. (But A ¡ B 6= B ¡ A). { Associative: (A \ B) \ C = A \ (B \ C) = A \ B \ C. (also for [) { Distributive: A \ (B [ C) = (A \ B) [ (A \ C) A [ (B \ C) = (A [ B) \ (A [ C) { DeMorgan's laws: (A \ B)c = Ac [ Bc (A [ B)c = Ac \ Bc ES150 { Harvard SEAS 3 Elements of probability theory A probabilistic model includes ² The sample space of an experiment { set of all possible outcomes { ¯nite or in¯nite { discrete or continuous { possibly multi-dimensional ² An event A is a set of outcomes { a subset of the sample space, A ½ . -
Sample Space, Events and Probability
Sample Space, Events and Probability Sample Space and Events There are lots of phenomena in nature, like tossing a coin or tossing a die, whose outcomes cannot be predicted with certainty in advance, but the set of all the possible outcomes is known. These are what we call random phenomena or random experiments. Probability theory is concerned with such random phenomena or random experiments. Consider a random experiment. The set of all the possible outcomes is called the sample space of the experiment and is usually denoted by S. Any subset E of the sample space S is called an event. Here are some examples. Example 1 Tossing a coin. The sample space is S = fH; T g. E = fHg is an event. Example 2 Tossing a die. The sample space is S = f1; 2; 3; 4; 5; 6g. E = f2; 4; 6g is an event, which can be described in words as "the number is even". Example 3 Tossing a coin twice. The sample space is S = fHH;HT;TH;TT g. E = fHH; HT g is an event, which can be described in words as "the first toss results in a Heads. Example 4 Tossing a die twice. The sample space is S = f(i; j): i; j = 1; 2;:::; 6g, which contains 36 elements. "The sum of the results of the two toss is equal to 10" is an event. Example 5 Choosing a point from the interval (0; 1). The sample space is S = (0; 1). E = (1=3; 1=2) is an event. Example 6 Measuring the lifetime of a lightbulb. -
PA As a Stochastic Process
Anti-Essentialism in Public Administration Conference A decentering tendency has undermined the foundations of public administration theory Fort Lauderdale, FL March 2-3, 2007 Theorizing Public Administration as a Stochastic Process Catherine Horiuchi University of San Francisco Working Paper: Not for attribution or citation with author’s permission Abstract The capacity of information networks to capture and manipulate ever-larger streams of globally acquired, real-time data accelerates fragmentation of traditional public administration protocols, away from managing stable states toward temporary and permeable framing of governmental and corporate interests. Technologies simultaneously provide historic opportunities for dissent, individualism, and small-d democratic movements. Intermittent, overlapping governance – characterized by private government, small wars, state failures, and opportunistic shifts of power and capital from public stewardship to private parties – results in ideological or pragmatic retreats from and progressions of institutional boundaries. Ephemeral balances rather than negotiated long-term settlements demark the edges of public and private. This fluidity of realms increasingly affects the duration and allocation of administrative responsibilities between formerly firmly edged divisions of local, state, and national governments. The assumption of a static state in public administration theory does not hold. Government becomes a metaphorical fluid, an eddy that retains its shape more or less, long enough to become an object of analysis and action. The new assumption of administrative fluidity invokes a world of measurement estimating impacts of partially ordered and partially stochastic events. Sensemaking derives from sophisticated evaluative and probabilistic analyses. Traditional construction of the field with its assumption of durable governmental operations may no longer be a best-fit theory for multi-layered, ephemeral states. -
Quantum Mechanics and Free Will: Counter−Arguments
Quantum mechanics and free will: counter−arguments Martín López−Corredoira Astronomisches Institut der Universitaet Basel, Venusstrasse 7, CH−4102−Binningen, Switzerland E−mail: [email protected] Abstract Since quantum mechanics (QM) was formulated, many voices have claimed this to be the basis of free will in the human beings. Basically, they argue that free will is possible because there is an ontological indeterminism in the natural laws, and that the mind is responsible for the wave function collapse of matter, which leads to a choice among the different possibilities for the body. However, I defend the opposite thesis, that free will cannot be defended in terms of QM. First, because indeterminism does not imply free will, it is merely a necessary condition but not enough to defend it. Second, because all considerations about an autonomous mind sending orders to the body is against our scientific knowledge about human beings; in particular, neither neurological nor evolutionary theory can admit dualism. The quantum theory of measurement can be interpreted without the intervention of human minds, but other fields of science cannot contemplate the mentalist scenario, so it is concluded that QM has nothing to say about the mind or free will, and its scientific explanation is more related to biology than to physics. A fatalistic or materialist view, which denies the possibility of a free will, makes much more sense in scientific terms. 1. Definition of free will First of all, we must clarify the meaning of the term to which we refer: Free will: a source totally detached from matter (detached from nature) which is the origin (cause) of options, thoughts, feelings,.. -
Probability Spaces Lecturer: Dr
EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 4: Probability Spaces Lecturer: Dr. Krishna Jagannathan Scribe: Jainam Doshi, Arjun Nadh and Ajay M 4.1 Introduction Just as a point is not defined in elementary geometry, probability theory begins with two entities that are not defined. These undefined entities are a Random Experiment and its Outcome: These two concepts are to be understood intuitively, as suggested by their respective English meanings. We use these undefined terms to define other entities. Definition 4.1 The Sample Space Ω of a random experiment is the set of all possible outcomes of a random experiment. An outcome (or elementary outcome) of the random experiment is usually denoted by !: Thus, when a random experiment is performed, the outcome ! 2 Ω is picked by the Goddess of Chance or Mother Nature or your favourite genie. Note that the sample space Ω can be finite or infinite. Indeed, depending on the cardinality of Ω, it can be classified as follows: 1. Finite sample space 2. Countably infinite sample space 3. Uncountable sample space It is imperative to note that for a given random experiment, its sample space is defined depending on what one is interested in observing as the outcome. We illustrate this using an example. Consider a person tossing a coin. This is a random experiment. Now consider the following three cases: • Suppose one is interested in knowing whether the toss produces a head or a tail, then the sample space is given by, Ω = fH; T g. Here, as there are only two possible outcomes, the sample space is said to be finite. -
Section 4.1 Experiments, Sample Spaces, and Events
Section 4.1 Experiments, Sample Spaces, and Events Experiments An experiment is an activity with specified observable results. Outcomes An outcome is a possible result of an experiment. Before discussing sample spaces and events, we need to discuss sets, elements, and subsets. Sets A set is a well-defined collection of objects. Elements The objects inside of a set are called elements of the set. a 2 A ( a is an element of A ) a2 = A ( a is not an element of A) Roster Notation for a Set Roster notation will be used exclusively in this class, and consists of listing the elements of a set in between curly braces. Subset If every element of a set A is also an element of a set B, then we say that A is a subset of B and we write A ⊆ B. Sample Spaces and Events A sample space, S, is the set consisting of ALL possible outcomes of an experiment. Therefore, the outcomes of an experiment are the elements of the set S, the sample space. An event, E, is a subset of the sample space. The subsets with just one outcome/element of the sample space are called simple events. The event ? is called the empty set, and represents the event where nothing happens. 1. An experiment consists of tossing a coin and observing the side that lands up and then rolling a fair 4-sided die and observing the number rolled. Let H and T represent heads and tails respectively. (a) Describe the sample space S corresponding to this experiment. -
What Psychodynamic Psychotherapists Think About Free Will and Determinism and How That Impacts Their Clinical Practice : a Qualitative Study
Smith ScholarWorks Theses, Dissertations, and Projects 2012 What psychodynamic psychotherapists think about free will and determinism and how that impacts their clinical practice : a qualitative study Patrick J. Cody Smith College Follow this and additional works at: https://scholarworks.smith.edu/theses Part of the Social and Behavioral Sciences Commons Recommended Citation Cody, Patrick J., "What psychodynamic psychotherapists think about free will and determinism and how that impacts their clinical practice : a qualitative study" (2012). Masters Thesis, Smith College, Northampton, MA. https://scholarworks.smith.edu/theses/872 This Masters Thesis has been accepted for inclusion in Theses, Dissertations, and Projects by an authorized administrator of Smith ScholarWorks. For more information, please contact [email protected]. Patrick Cody What Psychodynamic Psychotherapists Think about Free Will and Determinism and How that Impacts their Clinical Practice: A Qualitative Study ABSTRACT This qualitative study explored psychodynamic psychotherapists’ beliefs about free will and determinism and how these impact their work with clients. A secondary goal was to determine if and how knowledge of psychodynamic theory, neuropsychology and/or physics has shaped those views. Twelve clinicians were asked questions related to free will, determinism and clients’ behavioral change. All participants said that psychodynamic theory has influenced their beliefs, and a majority said that neuropsychology has done so. Major findings include that 11 of the 12 participants endorsed the concept of compatibilism, that free will and determinism can co- exist and are not mutually exclusive in impacting behavior. This finding compares to, but does not confirm, research that found psychodynamic clinicians were more deterministic than other clinicians (McGovern, 1986), and it contrasts with research that suggests that the science related to free will and determinism has not reached the field and influenced clinical practice (Wilks, 2003). -
7.1 Sample Space, Events, Probability
7.17.1 SampleSample space,space, events,events, probabilityprobability • In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas. BlaiseBlaise PascalPascal--fatherfather ofof modernmodern probabilityprobability http://www-gap.dcs.st- and.ac.uk/~history/Mathematicians/Pascal.html • Blaise Pascal • Born: 19 June 1623 in Clermont (now Clermont-Ferrand), Auvergne, France Died: 19 Aug 1662 in Paris, France • In correspondence with Fermat he laid the foundation for the theory of probability. This correspondence consisted of five letters and occurred in the summer of 1654. They considered the dice problem, already studied by Cardan, and the problem of points also considered by Cardan and, around the same time, Pacioli and Tartaglia. The dice problem asks how many times one must throw a pair of dice before one expects a double six while the problem of points asks how to divide the stakes if a game of dice is incomplete. They solved the problem of points for a two player game but did not develop powerful enough mathematical methods to solve it for three or more players. PascalPascal ProbabilityProbability • 1. Important in inferential statistics, a branch of statistics that relies on sample information to make decisions about a population. • 2. Used to make decisions in the face of uncertainty. TerminologyTerminology •1. Random experiment : is a process or activity which produces a number of possible outcomes. The outcomes which result cannot be predicted with absolute certainty. • Example 1: Flip two coins and observe the possible outcomes of heads and tails ExamplesExamples •2. -
Probabilities of Outcomes and Events
Probabilities of outcomes and events Outcomes and events Probabilities are functions of events, where events can arise from one or more outcomes of an experiment or observation. To use an example from DNA sequences, if we pick a random location of DNA, the possible outcomes to observe are the letters A, C, G, T. The set of all possible outcomes is S = fA, C, G, Tg. The set of all possible outcomes is also called the sample space. Events of interest might be E1 = an A is observed, or E2 = a purine is observed. We can think of an event as a subset of the set of all possible outcomes. For example, E2 can also be described as fA; Gg. A more traditional example to use is with dice. For one roll of a die, the possible outcomes are S = f1; 2; 3; 4; 5; 6g. Events can be any subset of these. For example, the following are events for this sample space: • E1 = a 1 is rolled • E2 = a 2 is rolled • Ei = an i is rolled, where i is a particular value in f1,2,3,4,5,6g • E = an even number is rolled = f2; 4; 6g • D = an odd number is rolled = f1; 3; 5g • F = a prime number is rolled = f2; 3; 5g • G = f1; 5g where event G is the event that either a 1 or a 5 is rolled. Often events have natural descriptions, like \an odd number is rolled", but events can also be arbitrary subsets like G. Sample spaces and events can also involve more complicated descriptions.