What is Probability theory: Definition and 101 Discussions
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.
Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.
Some time ago we had a discussion of the sleeping beauty problem
https://www.physicsforums.com/threads/the-sleeping-beauty-problem-any-halfers-here.916459/
which is a well known problem in probability theory. In that thread, there was no consensus whether the probability of heads is 1/2 or 1/3...
A recent question about interpretations of probability nicely clarified the role of the Kolmogorov axioms:
[... some excursions into QM, negative probabilities, and quasiprobability distributions ...]
Conclusion: the Kolmogorov axioms formalize the concept of probability. They achieve this by...
Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable. Is there a reason why the choice is also preferred in physics?
For concretness I'll use atoms and photons but this problem is actually just about probabilities.
There's an atom A whose probability to emit a photon between times t and t+dt is given by a gaussian distribution probability P_A centered around time T_A with variance V_A. There's a similar atom...
I am interested in stomach acid and heat expansion, for instance the stomach will become heated due to an athelete competing. The heat causes atheletes to live shorter than people who don't have their body heated so often. I do a lot of differential equations and number theory, but I was...
I submitted this solution, and it was marked incorrect. Could I get some feedback on where I went wrong?
Let S represent the event that Party A wins the senate and H represent the event that Party A wins the house.
There are 4 cases: winning the senate and house (##S \cap H##), winning just...
I got (a) and (b) but I'm still working on (c). The solutions can be found here for your reference: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/unit-ii/lecture-9/MIT6_041SCF13_assn05_sol.pdf. But...
I tried to derive the right hand side of the Radon-Nikodym derivative above but I got different result, here is my attempt:
\begin{equation} \label{eq1}
\begin{split}
\frac{\mathrm d\mu_{\Theta\mid X}}{\mathrm d\mu_\Theta}(\theta \mid x) &= f_{\Theta\mid X}(\theta\mid x) \mathrm \space...
Because I do have a background in the latter it was originally very difficult for me to understand some aspects of QP (quantum physics) when I initially learned it. More specifically whenever probabilities were involved I couldn’t really make full sense of it while I never had any problems...
Homework Statement
Let ##U_1, U_2, U_3## be independent uniform on ##[0,1]##.
a) Find the joint density function of ##U_{(1)}, U_{(2)}, U_{(3)}##.
b) The locations of three gas stations are independently and randomly placed along a mile of highway. What is the probability that no two gas...
Homework Statement
Hi all, I have some difficulty understanding the following problem, help is greatly appreciated!
Let ##U_1, U_2, U_3## be independent random variables uniform on ##[0,1]##. Find the probability that the roots of the quadratic ##U_1 x^2 + U_2 x + U3## are real.
Homework...
Homework Statement
[/B]
Hi all, I have an issue understanding the concepts pertaining to the following problem, assistance is greatly appreciated.
I understand the "flow" of the problem; first find the probability of obtaining balls of the same colour, then use the geometric distribution...
Homework Statement
Discrete random variables ##X,Y,Z## are mutually independent if for all ##x_i, y_j, z_k##,
$$P(X=x_i \wedge Y=y_j \wedge Z=z_k ) = P(X=x_i)P(Y=y_j)P(Z=z_k )$$
I am trying to show (or trying to understand how someone has shown) that ##X,Y## are also independent as a result...
Homework Statement
Hi all, could someone give my working a quick skim to see if it checks out? Many thanks in advance.
Suppose that 5 cards are dealt from a 52-card deck. What is the probability of drawing at least two kings given that there is at least one king?Homework Equations
The Attempt...
<Moderator's note: Moved from homework.>
Hi all, I have an issue understanding a statement I read in my text.
It first states the following Proposition (Let's call it Proposition A):
The number of unordered samples of ##r## objects selected from ##n## objects without replacement is ##n...
Suppose I am trying to approximate a function which I do not know, but I can measure. Each measurement takes a lot of effort.
Say the function I am approximating is ##y=f(x)## and ##x \in [0,100]##
Supose I know the expectation and variance of ##f(x)##.
Is there a way to compute the confidence...
Hi everyone.
As a graduate student in statistics, I had taken a graduate course in measure-theoretic probability theory. In a conversation with the professor, he had remarked that if I wanted to pursue further research on some of the topics covered, it may be wise to do background reading or...
Homework Statement
Homework Equations
All needed are in the picture above (i hope so)
The Attempt at a Solution
to me it is extremely difficult because it is so complicated with many notations. Also, I actually don't know how to read the question properly to answer it
Is E(beta) is the...
Homework Statement
The time (minute) that it takes for a terrain runner to get around a runway is a random variable X with the tightness function
fX = (125-x)/450 , 95≤x≤125
How big is the probability of eight different runners, whose times are independent after 100 minutes:
a) Everyone has...
Homework Statement
In a vessel is a 5 cent coin and two 1-cent coins. If someone takes up two randomly chosen of these coins, and we let X be the total value of the coins taken, what is the probability function for X?
Homework Equations
I know that X has a value {2,6}
The Attempt at a...
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...