Probability theory Definition and 129 Threads

Homework Statement Prove the continuity from below theorem. Homework EquationsThe Attempt at a Solution So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A. I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity) to...

Hello everyone! I'm studying the physics of complex systems and I'm approaching probability theory. I understand that we need a ## \sigma-algebra ## and the Kolmogorov axioms in order to define a probability space but then I bumped into the following relation: $$ p(A_1 \cup A_2 ) = p( A_1 ) + p(...

I study control theory and robotics. Recently I figured out that I have a much deeper understanding of probability and statistics compared to my colleagues. Is this 'talent' valuable in my field and if so, where? We used this theory to define white noise, but nothing more...as of now. Also I am...

Hi All This is in relation to the folllowing paper: https://arxiv.org/pdf/1402.6562.pdf See section 3 on examples where standard probability theory is discussed. Is it valid? To me its rather obvious but I had had a retired professor of probability say probability theory doesn't have a state...

Su, Francis, et. al. have a short description of the paradox here: https://www.math.hmc.edu/funfacts/ffiles/20001.6-8.shtmlI used that link because it concisely sets forth the paradox both in the basic setting but also given the version where the two envelopes contain ( \,\$2^k, \$2^{k+1}) \...

Would you please introduce me some textbooks that offer multiple choice or true-false questions at the end of each chapter so I can practice before my final exam? Basically, that class I am taking right now cover many central topics in Undergraduate Probability course such as: • Moments and...

In another math thread https://www.physicsforums.com/threads/categorizing-math.889809/ several people expressed their opinion that, while statistics is a branch of applied mathematics, the probability theory is pure mathematics and a branch of analysis, or more precisely, a branch of measure...

Hi, I am looking for a book for studying probability theory using measure theory. This is the first course I am taking of probability. Notions and theorems from measure theory are part of this course. As it turns out, this is a catastrophic disaster, and the textbook for this course is also not...

Dear Physics Forum personnel, I would like to seek your recommendation on a good, introductory textbook in the probability theory (non measure-theoretic treatment) that contains both the applied and theoretical treatment of the subject. My goal is to advance into the measure-theoretic...

Homework Statement 3. The Attempt at a Solution [/B] ***************************************** Can anyone possibly explain step 3 and 4 in this solution?

Homework Statement Calculate the limit $$lim_{s,t→∞} R_X(s, s+t) = lim_{s,t→∞}E(X(s)X(s+t))$$ for a continuous time Markov chain $$(X(t) ; t ≥ 0)$$ with state space S and generator G given by $$S = (0, 1)$$ $$ G= \begin{pmatrix} -\alpha & \alpha \\ \beta & -\beta\...

Homework Statement Let $$(X(n), n ∈ [1, 2])$$ be a stationary zero-mean Gaussian process with autocorrelation function $$R_X(0) = 1; R_X(+-1) = \rho$$ for a constant ρ ∈ [−1, 1]. Show that for each x ∈ R it holds that $$max_{n∈[1,2]} P(X(n) > x) ≤ P (max_{n∈[1,2]} X(n) > x)$$ Are there any...

Homework Statement Let X(t) be a birth-death process with parameters $$\lambda_n = \lambda > 0 , \mu_n = \mu > 0,$$ where $$\lambda > \mu , X(0) = 0$$ Show that the total time T_i spent in state i is $$exp(\lambda−\mu)-distributed$$ 3. Solution I have a hard time understanding this...

This question is killing me. I know the graph is non-monotonic so i have to split up finding F(Y) for -1<Y and Y<1 but then what do I do with the modulus? >.< Any help would be greatly appreciated! Thank you so much x

Homework Statement An only child Urška puts 3 pieces of paper into a bag : on each piece of paper is written a name of one of her classmates :David,Niko,Dejan (those are the 3 names): She then randomly picks 2 pieces of paper from the bag and checks them Match the statements on the right with...

Prove or disprove the following statement: If p(a)=p(b)=q then p(a∩b)≤q2 We know nothing know about event a , b. The Attempt at a Solution I tried this but don't know correct or not Can some one help me let a, b are independent event 0<q<1 then p(a∩b) = p(a) p(b) = q*q = q^2 [/B]

For three dice, you can have 6 * 6 * 6 = 216 permutations (order matters). The dice has a uniform probability distribution of p(x) = 1/6. Easy. I'm trying to get an estimate of how many permutations you can have if a variable has a normal probability distribution. So for example, if a...

I need another class for a 6 week summer semester and I'm curious if probability theory is generally a class you wouldn't want to cram in 6 weeks with another class? The only college level probability I've done was in a discrete math course but I'm fine with other areas since I also took...

Hello, I am analysing hydrology data and curve fitting to check the best probability distribution among 8 candidate distribution. (2 and 3 parameter distributions) The selection is based on the lowest AIC value. While doing my calculation in excel, how is it suggested to treat very low (approx...

I am think what is the structure of generated ##\sigma##-algebra. Let me make it specific. How to represent ##\sigma(\mathscr{A})##, where ##\mathscr{A}## is an algebra. Can I use the elements of ##\mathscr{A}## to represent the element in ##\sigma(\mathscr{A})##?

Suppose ##\mu:\mathcal{F}\rightarrow[0,\infty)## be a countable additive measure on a ##\sigma##-algebra ##\mathcal{F}## over a set ##\Omega##. Take any ##E\subseteq \Omega##. Let ##\mathcal{F}_{E}:=\sigma(\mathcal{F}\cup\{E\})##. Then, PROVE there is a countable additive measure...

Let x(a) be the extinction probability of a branching process whose offspring is Poisson distributed with parameter a. I need to find the limit as a approaches infinity x(a)e^a. I tried computing x(a) directly using generating functions, and I found that it's the solution to e^(a(s-1))=s, but...

Homework Statement Alice attends a small college in which each class meets only once a week. She is deciding between 30 non-overlapping classes. There are 6 classes to choose from for each day of the week, Monday through Friday. Trusting in the benevolence of randomness, Alice decides to...

I'm a grad student studying electrical/computer engineering. Since I have a month of winter break coming up soon, I want to use it to study some more about probability theory and stochastic processes. Has anyone previously done a self study or partner study over a break like this? If so, how did...

As I realized recently, the probability theory as used in quantum mechanics does not follow Kolmogorov's axioms. I am interested in a book that treats probability theory as it is done in quantum mechanics. Is this treated in books on quantum logic? Any other good book on the mathematical...

Q1. There are n cells and each cell contains k balls. One ball is taken from each of the cells. Find the probability that the second lowest label from the balls drawn is m. Q2. Game played by two friends: each player picks two balls. The person who gets the first white ball in the second draw...

I think the first thing that is confusing me is the terminology. There are too many similar terms (e.g. probability measure, probability distribution, probability density function, probability mass function) What are the general concepts and what are the instances of those concepts? Like, are...

Homework Statement Five cards numbered 1 to 5 are shuffled and placed face down on a table. Two of the cards are picked at random. [Hint: find all of the possible outcomes of this experiment which form the sample space S and use the Equally Likely Principle.] Find the probability of the...

In probability theory a sample space is a set containing all possible outcomes of an experiment and an event is a subset of the sample space (an element of its power set). I think it would be natural to think of the basis of the vector space representing a quantum system as a sample space and...

Homework Statement A random variable has a Poisson distribution with parameter λ = 2. Compute the following probabilities, giving an exact answer and a decimal approximation. P(X ≥ 4) Homework Equations P(X = k) = λke-λ/k! The Attempt at a Solution P(X ≥ 4) = Ʃk = 4∞...

Hello. I have a problem with probability theory task. The task is: X and Y is independent random variables with same density function fx=fy=f. What will be probability of P(X>Y). This P(X>Y) reminds me a cdf: P(X>Y)=1-P(X<Y)=1-cdf of X. Cdf of x is equal to integral ∫f dx from -inf to...

Hello. I have problem with probability theory task. Sorry for my english but i'll try to define the task. There are four classmates. Ana, Beta, Ceta and Deta. During the break all of them tiffed (probability is p) or became best friends (probability is 1-p). And all with each other tiffed or...

Author: William Feller Title: An Introduction to Probability Theory and Its Applications Amazon Link: https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20 https://www.amazon.com/dp/0471257095/?tag=pfamazon01-20 Prerequisities: Table of Contents for Volume I: Introduction: The Nature of...

Homework Statement Let U, V be random variables on [0,+\infty) with probability density functions f_U(x)=2e^{-2x} and f_V(x)=e^{-x}. 1. Give a coupling of U and V under which \{U\geq V\} with probability 1. 2. Give a maximal coupling of U and V. Homework Equations Cumulative distribution...

Hello, so suppose we have B(n,p), where n is discretely uniformly distributed on the integers of the interval (1,5) Is the expected value 3p, and is the variance 3p-p^2 ? I arrived at those answers by treating n as another variable, so np/5 summed over all n is 3p, and similar logic for...

Homework Statement Well, not really, but in essence that's the part I'm having trouble with. The actual question is The equality seems obvious enough, but I'm unsure how to actually prove it... Homework Equations N/A The Attempt at a Solution So it seems I have to prove that P( \{ E(Xh(Y)|Y)...

Let K be a standard normal random variable. Find the densities of each of the following random variables: X= |K| Y = K2 I get: fX(x) = √(2/π) e-x2/2 and fY(y) = 1/√(2*π) 1/√y e-y2/2

Homework Statement Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads 55 percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall...

reading this http://uk.arxiv.org/abs/1004.2529 about supposed parallels between the mathematical structure of probability in QM and some problems in economics question is that are there really any violations of classical probability theory, such as Pr(A) > Pr(A \cup B) in QM? The supposed...

Probability theory... It may make on sense but I want to understand probability intuitively...!...can anyone explain...?how should i start...?

As the title. Why are sigma fields important in probability? The only one reason I can think of is that sigma fields are used as domain, e.g. borel fields uses sigma fields instead of power set. However, are there any other significances of sigma fields in probability theory? Thanks for your...

Homework Statement EX=EY=5, VarX=1, VarY=sigma^2 >1 Z=aX+(1-a)Y, 0<=a<=1 find a that minimizes VarZ, and another a that maximize VarZ Homework Equations The Attempt at a Solution Not even sure where to begin *EX=5, VarX=1 thus EX^2 = 26 marginal px(x) =26/x^2 = 5/x but...

I find the probability theory I'm doing in college very difficult until I start wording it all out in my head. If I word it out then there's no confusion about what P(A) represents and what P(B|A) represents etc. but if I don't word it out then I have trouble thinking about it. I think its the...

Homework Statement [/b] There are two problems I need help with, which are posted below. Any help is appreciated. 1)Let X have a Poisson distribution with parameter λ. If we know that P(X = 1|X ≤ 1) = 0.8, then what is the expectation and variance of X? 2)A random variable X is a sum of...

I have the choice of taking, next spring, a course on linear algebra (theoretical, covers linear operator theory) or a course on applied math (covers asymptotic analysis and pdes). I cannot take both because i would be taking too many classes at once, and the classes i am already planning on...

Homework Statement Discrete random variables X and Y , whose values are positive integers, have the joint probability mass function , (, ) = 2−−. Determine the marginal probability mass functions () and (). Are X and Y independent? Determine [], [ ], and [ ]. The Attempt at a Solution...

Here is a simple problem but with a lot of hidden difficulty: We have a weather station somewhere in the country, with the aid of satelitte information and it gives us the probability of rain every day. Suppose it is an average value p, for the season. At some other place there is an...

Homework Statement (Question is #6 on p.171 in An Introduction to Probability and Statistics by Ruhatgi & Saleh) Let X have PMF Pλ{X=x} = λxe-λ/x!, x=0,1,2... and suppose that λ is a realization of a RV Λ with PDF f(λ)=e-λ, λ>0. Find E(e-Λ|X=1) The Attempt at a Solution The...

Hi, I'm having some trouble solving one of the problems from my homework assignment. Homework Statement Prove: P((AB)C) = P(A(BC)) Where A,B,C are either true or false. Homework Equations We can't do this by using a truth table, we can use the following equations: P(A + B) = P(A) + P(B) -...

Homework Statement Let X be a geometric(p) random variable for some p ∈ (0, 1). For n, k ∈ N, show that P(X = n + k |X > n) = P(X = k). Now, explain this property using the interpretation of X as the first successful trial among independent trials each of probability p. Homework...