In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy.
The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.
Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, particularly in the field of computational science. By analogy, quasi-Monte Carlo methods use quasi-random number generators.
Random selection, when narrowly associated with a simple random sample, is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, say research subjects, has the same probability of being chosen, then we can say the selection process is random.According to Ramsey theory, pure randomness is impossible, especially for large structures. Mathematician Theodore Motzkin suggested that "while disorder is more probable in general, complete disorder is impossible". Misunderstanding this can lead to numerous conspiracy theories. Cristian S. Calude stated that "given the impossibility of true randomness, the effort is directed towards studying degrees of randomness". It can be proven that there is infinite hierarchy (in terms of quality or strength) of forms of randomness.
It has been awhile since I took Physics. The other day my daughter asked some questions about force. She told her teacher explained if the object is moving horizontal the push force is 0. I said that can't be right. There has to be a force to overcome friction. This lead me thinking...
Homework Statement
Given an array with random real elements, find the indices pair (i, j) where j>i s.t
a[j] - a[i] is maximised in O(n ln n) time
I tried sorting the array of (value, index) pair with O(n ln n) sort, and then inspect the index array. However it does not suggest me any...
Homework Statement
1.
Suppose u flip a coin
Z = 1 if the coin is heads
Z = 3 if the coin is tails
W = Z^2 + Z
a)
what is the probability function of Z?
b)
what is the probability function of W?
2.
Let Z ~ Geometric (theta). Compute P(5<=Z<=9).
Homework Equations
The Attempt at a Solution...
Homework Statement
A clown stands at the side of a swimming pool. In his hand is a bag containing n red balls and n blue balls. At each step he puts his hand into the bag and pulls out a random ball and throws it away. If the ball is red, he makes a step towards the pool and if it is blue...
Well, I thought I understood the difference between (weak) convergence in probability, and almost sure convergence.
My prof stated that when dealing with discrete probability spaces, both forms of convergence are the same.
That is, not only does A.S. convergence imply weak convergence, as...
I've read a book named "Cheating Time : Science, Sex and Aging" by Roger Gosden. There he mentions that natural selection occurs as a process of mutations. So suppose we have some individuals of a species that have suddenly got some mutation which gives them better chance of survival in a given...
Homework Statement
Suppose X1,X2... are iid mean 1 exponential random variables. Use large deviation methodology to give a lower bound for the rate function R(a) for a>1
Homework Equations
R(a) \leq \frac{-logP[Sn >n*a]}{n}
The Attempt at a Solution
I know that a sum of exponential random...
Hi all, this has been bugging me so maybe you can help me out.
In the front window of my house, there is a dent in the glass that looks somewhat similar to this:
http://www.flickr.com/photos/guerra/2557095422/"
However, there is not hole in the glass. But the shape of the dent is...
Hi
In http://sl-proj-bi-specification.web.cern.ch/sl-proj-bi-specification/Activities/Glossary/techglos.pdf it says that: ... if the sources of uncertainties are numerous, the Gaussian distribution is generally a good approximation.
I don't quite understand why. The Central Limit Theorem (CLT)...
The transformation of a random variable is well documented and there are numerous examples on the web. Most examples present univariate variable transformation utilising inverse of the transformation function. The method works whenever the transformation function is one-to-one.
Let's say...
Comment made by an 11 year old accompanied by her grandmother after she entered the Subway I was eating at, somewhere in The Bronx, NY:
"Why would I want to learn math? Just so I can work at a cash register?"
I wanted to cry...:frown:
Where would she get that idea from?
I am trying to linearize a function, f(x), where x is a normally distributed N(0,1) random variable. How can I perform a taylor series expansion around a deterministic value x0? Thanks.
What is the motivation behind random variables in probability theory?
The definition is easy to understand. Given a probability space (Ω, μ), a random variable on that space is an integrable function X:Ω→R. So essentially, it allows you to work in the concrete representation R instead of the...
The only requirement is that the pure random numbers generated have to be purely random to its creator, not for some other inferior species.
(To keep Evo happy, a super being is someone who is omniscient and omnipotent.)
Homework Statement
Continuous random variable X has probability density function defined as
f(x)= 1/4 , -1<x<3
=0 , otherwise
Continuous random variable Y is defined by Y=X^2
Find G(y), the cummulative distribution function of Y
Homework Equations
The Attempt at a...
Ok I am not sure if I should put this question in the homework category of here but it’s a problem from schaums outline and I know the solution to it but I don’t understand the solution 100% so maybe someone can explain this to me.
Let X and Y be defined by:
\begin{array}{l}
X = \cos \theta...
If X is uniformly distributed over [0,a), and Y is independent, then X + Y (mod a) is uniformly distributed over [0,a), independent of the distribution of Y.
Can anyone point me to a statistics text that shows this?
Thanks,
1 Calculate the expected value of variable x (or E(x)) (number of trials * probability of success)
2 Calculate the variance (expected value * probability of a failure)
Take everything to the right of the decimal in the variance off. Then the range of future values is E(x) plus/minus the variance.
Hi I'm making a program that generates a different arrangement of letters in C.
For example, the word ALTEC will be rearranged randomly by the program.
the output would be "LETAC" or "CEALT" etc.
The program should use all the given letters and not repeat the letters.
Here's a code...
Please help - I am helping a student who is trying to translate Ong-Schnorr to C++ or VB.net language. My major was electronics so I am not sure how to interpret inverse random integer k. Any one ? or suggest me a reference to read on - much thanks, newbie
Ong-Schnorr-Shamir (from Briuce's...
Hi,
I'm modeling receptors moving along a cell surface that interact with proteins inside of a cell. I figured it would be easier to model the receptors in spherical coordinates, however I'm unsure of how to model a random walk. In cartesian coordinates, I basically model a step as:
x = x +...
Hi friends/colleagues,
Let X1, X2, ..., Xn be a sequence of independent, but NOT identically distributed random variables, with E(Xi)=0, and variance of each Xi being UNEQUAL but finite.
Let S be the vector of partial sum of Xs: Si=X1+X2+...+Xi.
Question: What is the limiting...
Homework Statement
A random variable X takes values 1,2,...,n with equal probabilities. Determine the expectation, R for X and show that the variance, Q^2 is given by 12Q^2=n^2-1. Hence, find
P(|X-R|>Q) in the case n=100
Homework Equations
The Attempt at a Solution
I can show...
Given two random variables x and y, and a constant c
What conditions are needed to make:
Prob( w x + y < c ) \approx Prob( w x < c ), \text{ for } w \rightarrow \infty
Can anyone help? I think E(x) < \infty and E(y) < \infty might do. Is this right?
tks!
Just a random thought
Would anyone like to share some dreaming experiences?
My dreams seem very vivid-I can remember them in full color and detail, and conscious reasoning is possible in my dreams.
However, there are times when my body moves (or I sense that my body moves/my line of...
Hello,
I am trying to find a clean solution for this (i.e. straightforward and academically acceptable). I appreciate if someone can help me.
1- I need to select between selections 1 to n (in my case 1-8 ). However each preceding item should have higher probability than the next one. For...
Just to help define what random is Dictionary.com states that it is:
"–adjective
1. proceeding, made, or occurring without definite aim, reason, or pattern: the random selection of numbers."
But does random truly exist?
For example the roll of dice is usually referred to as random, but...
Homework Statement
Say, it is known that
E_X[f(X)] = E_X[g(X)] = a where f(X) and g(X) are two functions of the same random variable X. What is the relationship between f(X) and g(X)?
Homework Equations
The Attempt at a Solution
My answer is f(X) = g(X) + h(X) where E_X[h(X)] =...
Homework Statement
Let U = 0.X1X2X3... be a random number in (0,1].
1) Find the distribution of every decimal digit Xi, i = 0,1,2...
2) Show that they are independent of each other
The Attempt at a Solution
I could use a hint for N°2. I have an idea, but I think it's wrong...
Homework Statement
Hello,
have a stats question I am hoping you guys can help with. The expectation of a function g of a random variable X is:
E[g(X)] = \int^{\infty}_{-\infty} g(x)fx(x)dx
where fx is the pdf of X. For example, the particular expectation I am considering right now...
Homework Statement
Let X and Y be i.i.d normal random variables with mean 0 and variance (that is, N(0,1)). If Z=min(X,Y). Prove that the square of Z is a Gamma distribution and identify the parameters.
My problem is that the cdf of a normal random variable has no exact form. I need the cdf...
Hi, I'm trying to get it so that when the program initiates, if the random number is greater than 3, I turn left. here is what I have in summary;
int rt=3;
Random ra = new Random();
for (int c = 6; c <12; ++c);
{
}
if(rt<ra)
{...
This is a problem from my A levels Stats2 book. I understood the problem but one of my answers doesn't seem to be correct according to the book so I thought I better be sure!
Homework Statement
A piece of laminated plywood consists of 3 pieces of wood of type A and 2 pieces of type B. The...
Hello!
I've been trying to create an algorithm which picks a number randomly from array. P.S i have an array like A={1,4,-1,3,-7,2,-14} and I want to pick a number randomly from array.
Any idea?
Thank you.
If 2 random variables X, and Y have the same distribution, does that mean that for another random variables Z, X + Z and Y + Z also have the same distribution?
From looking at the convolution formula, the answer should be yes, because the convolution of random variables depends only on the...
Hello all,
I have the following question:
Assume (\Omega, \mathcal{F},P) = ([0,1],\mathcal{B}([0,1]),\lambda), where \lambda is Lebesgue mesure, so is X(\omega) = \frac{1}{\omega} a random variable defined on this probability space?
If yes, then can I say that X is bounded a.s. because the...
Hi I am currently reading:
http://www.cs.cornell.edu/~asaxena/learningdepth/saxena_ijcv07_learningdepth.pdf
which deals with reconstructing depth from a single still image.
A gaussian multiscale markov random field is trained in a supervised context where the model is shown below...
Hi all,
I have a question of computing the expectation of random sums.
E(sim_{k=1}^N X_k) = E(N)E(X) if N and X_1, X_2,...are independent and X_k's are iid. Here both N and X_k's are r.vs.
But the condition of N and X_1, X_2,...being independent is not true in many cases.
How will...
Dear Friends,
I want to find an upper and lower bound for the expected value of the minimum of independent binomial random variables. What paper/book do you suggest for this problem?
In other words, I need to find bounds for E(min(X1,X2,...,Xn)), where Xi 's are independent random variables...
Homework Statement
A car moving at constant acceleration covers the distance between two points 58m apart in 6.2 seconds. its speed as it passes the second point is 54 km/hr. what is the speed at the first point? at what prior distance from the first point was the car at rest?Homework Equations...
I am working on correcting an exam so that I may study for my probability final. Unfortunately, I don't have the correct answers, so I was hoping that someone here might be able to check my thought process.
1) Pick three numbers without replacement from the set {1,2,3,3,4,4,4}. Let T be the...
Re
www.physicsforums.com/showthread.php?t=403487
Actually, there is a move back to writing multiplication left to right. And while it is not wrong that this purely a matter of convention and symmetric within a narrow context, in a wider context, this notational choice acquires very real...
There are no Bayesian Networks for continuous random variables, as far as I know. And the Netica Bayesian Network software discretize continuous random variables to build bayesian models. Are there any reasons for this? Has anyone proposed continuous random variable bayesian networks?
Homework Statement
I'm trying to revise for a probability exam in a few weeks and still getting really confused :( so I'd be really grateful for a push in the right direction:
Consider a random walker on f0; 1; 2g who moves as follows:
- if at 0, probability 1/2 of staying at 0 and 1/2...
Hello,
I have read the probability chapter in Feynman's lectures on physics. And got fascinated by the random walk. There is a statement, that in a game where either a vertical distance of +1 or -1 can be walked each move, the expected value of the absolute distance (lets call it <D>) from...
Homework Statement
Problem 1 – Normal Random Variables
B) Y ~ N(300, 100). Pr (300 < Y < 320) = 0.4772
D) H ~ N(4000, 25). R = f(H) = 0.5H – 60. E(R) = 1940; Var(R) = 156.25I have a problem solving these problems above...I missed the class when we covered this subject and now I am lost...
Homework Statement
Given the joint density, f(x,y), derive the probability density function for Z = X + Y and V = Y - X.
Homework Equations
f(x,y) = 2 for 0 < x < y < 1
f(x,y) = 0 otherwise.
The Attempt at a Solution
For Z = X + Y, I can derive the fact that,
f_Z(z) =...
I am having a lot of trouble conceptually understanding the idea of a random effect in ANOVA testing - more specifically identifying whether a factor is random or fixed
Thanks,
Thrillhouse86
Homework Statement
Given 2 independent uniform random variables X,Y = U [0,1], consider the random variables Z = g (X,Y) for g = (x,y) = sqrt (-2ln(x) . cos(2piy). Since finding the distribution of g(X,Y) analytically is quite tough, I need to generate MATLAB program for
1 - 10,000...