What is Variables: Definition and 1000 Discussions
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol.
In computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context.
A bound variable is a variable that was previously free, but has been bound to a specific value or set of values called domain of discourse or universe. For example, the variable x becomes a bound variable when we write:
For all x, (x + 1)2 = x2 + 2x + 1.
or
There exists x such that x2 = 2.
In either of these propositions, it does not matter logically whether x or some other letter is used. However, it could be confusing to use the same letter again elsewhere in some compound proposition. That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae.
The term "dummy variable" is also sometimes used for a bound variable (more often in general mathematics than in computer science), but that use can create an ambiguity with the definition of dummy variables in regression analysis.
So I've never really understood this idea of a light-cone. I know there is literature, but what are the most important physical concepts/consequences about this light-cone model of special relativity?
Also, could you give a concise explanation of light-cone variables?
Realised I probably posted this in the wrong forum before, should've been here..
I often see a function's differential expressed in terms of convenient partial derivatives eg
dU=(dU/dT) dT + (dU/dV) dV
And I've seen it written that "any system is uniquely specified by two parameters, such...
Homework Statement
When possible express the general solution in explicit form.
Solve dy/dx =x^2 /(1+y^2)
Homework Equations
This is a first order non-linear ordinary differential equation.
The Attempt at a Solution
dy(1+y^2) = x^2 dx
Integration both sides returns:
y+ (y^3 )/3=...
Solve each differential equation. Express the general solution in explicit form.
y' = (3x^2 -1) / (3+2y)
So, I will skip many steps, because they are easy. However, I am stuck in one of the last ones.
y^2 +3y = x^3- x +C
y(y+3)= x^3 - x +C
I have seen the solution for y...
Homework Statement
[2 Dimensional infinite square well]
Show that you can separate variables such that the solution to the time independent schrodinger equation, ## \hat{H} \psi (x,y) = E \psi (x,y) ## can be written as a product state ## \psi (x,y) = \phi (x) \chi (y) ## where ## \phi (x)##...
Hi!
I was wondering: is it possible to have a non-orientable surface in 3D which is parametrized by u and v, with u and v periodic (i.e. is it possible to map the torus continuously into a non-orientable surface in 3D?)
If so, does anyone have any explicit examples?
Homework Statement
Determine if
f(x,y) = ((x-y)4 +x3 +xy2)/(x2+y2)
[f(x,y = 0 @ (0,0)]
is differentiable at the origin.
Homework Equations
x = (0,0)
The Attempt at a Solution
A function is differentiable at x if f(x+Δx) - f(x) = AΔx + |Δx|R(x)
Where A are constant...
Hi,
Homework Statement
The sample space of the following problem is defined thus: all the possible permutations of {1,2,3} including {1,1,1}, {2,2,2}, {3,3,3}. Suppose all results are equally probable. Let Xi denote the value of the ith coordinate, where i=1,2,3.
I am asked to determine...
The following is identically 0 which can be readily checked by a simple hand calculation.
$\binom{n+1}{k}2^{-n-1} - \binom{n}{k}2^{-n} + \binom{n}{k}*2^{-n-1} - \binom{n}{k-1}2^{-n-1}$
If you enter this in SAGE or Mathematica, using the appropriate script, and use full_simplify() and...
If we think of a functional as a function of the infinite number of taylor coefficients of the variable function, aren't they then just normal functions, a map between a set of reals to another set of reals.
ℂI am working on an assignment and have come across a question that I'm not quite sure how to approach. Here it is, with my "solution" and reasoning:
"[F]ind the limit at ∞ of the given function, or explain why it does not exist.
24. h(z) = Arg z , z \neq 0" (Complex Variables Second...
the random variables X1,X2... are independent and they take 0 and 1 values and they have expected value 0
if we have Y=X1+X2+...+Xn and Z=X1+X2+...+Xn+Xn+1 what is the ρ(Y,Z) for n=46
i know that ρ(Y,Z)=cov(Y,Z)/(sqrt(var(Y)*sqrt(var(Z)) but i need some help on how to find the cov and vars...
Hi,
Let's say I'm given X and Y identical independant continuous random variables.
We pose Z =X/Y, I remember there is a way to find the density function of Z, altough I can't get to remember how to do it and my probability book is out of town.(And I'm not so sure what to look for in google)...
Dear All,
If I have the linear algebraic system where its composed as of matrices in that form K*X=F, what column/row operations should I perform if I want to solve it where some of the X variables are known (targeted values) or if I want to solve when variables are constrained relative to...
we have X,Y variables and we have f(m,n)=P(X=m,Y=n)=C*((1/(a*m*n+15*m+11*n+8))^2)
for what value of a X,Y are indepedent? ( its not necessary to calculate C)
i know that f(m,n)=f(m)*f(n) if X,Y are independent but how i am going to use this for calculate a?
Hi
I have a certain experiment that I repeat 40 times and get the result:
0.001 +/- 0.004.
Now I've repeated the experiment using a different method (so it is essentially a new experiment) and I get a new value:
-0.002 +/- 0.003
Now, is it true to say there is no statistically...
Greetings-
In trying to solve a thermal stress problem, I have encountered an inhomogeneous differential equation of the following general form:
\nabla^2 \Phi(r,z) = F_r(r)F_z(z)
Solving the homogeneous case is no problem, as it is kind of a classic. Is there a route to finding a particular...
Dyadic Cube C_{k,N} = X \in\ \mathbb{R}^{n} \frac{k_{i}}{2^{N}} \leq x_{i} < \frac{k_{i}+1}{2^{N}} for 1 \leq i \leq n
Where k = \pmatrix {
k_{1} \cr
k_{2} \cr
\vdots \cr
k_{i} \cr
}
I understand that N is the level of the cubes, but what does k equal?
I'm having trouble...
Homework Statement
Let S be the part of the cylinder of radius 9 centered about z-axis and bounded
by y >= 0; z = -17; z = 17. Evaluate
\iint xy^2z^2
Homework Equations
The Attempt at a Solution
So I use the equation x^2 + y^2 \leq 9, meaning that r goes from 0 to 3
Since y...
Evaluate the limit or prove that it does not exist..
f(x,y) -> (0,0)
3xy/((x^2)+(4y^2))
The attempt at a solution:
Set x to 0 and you get 0
set y to 0 and you get 0
set y=x and you get 3x^2/5x^2 = 3/5
This means that limit does not exist.
Is this correct?
If this is correct, how...
we have X,Y variables and their common density f(m,n)=P(X=m,Y=n) where f(0,1)=0.1 f(1,0)=0.1
and f(1,1)=0,31 find P(X=0)
i think that P(X=0)= f(0,1) but it says that its incorrect what i am doing wrong?
Homework Statement
I can't see where the problem is in the following pseudo-code.
I recall from class that the problem is based on the signal calls and wait call in pickup() and putdown()
Thank you.
Homework Equations
class Monitor R()
{
bool forks[5]; // all true
condition c[5];
void...
I have a question regarding the paper by John Bell (www.drchinese.com/David/Bell_Compact.pdf ) in which he shows that a certain hidden variable approach cannot reproduce the expectation values predicted by QM for a pair of particles in the singlet state.
After eqn 15 on page 4, I don't...
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...
Let x(1),...,x(N) all be independent uniformally distributed variables defined on (0,1), i.e. (x(1),...,x(N)) - U(0,1). Define the random variable y(i) = x(i)/(x(1)+...+x(N)) for all i=1,...,N. I’m looking for the pdf of the random variables y(1),…,y(N). Has anyone come across such random...
Give a parametric equation for a curve in 2 dmiensions (x(t),y(t)) it may sometimes be possible to rotate or otherwise transform coordinates so that the tranformed curve becomes a function y = f(x) (as opposed to merely a relation such as x^2 = y^2 + 1). More generally, if we have a curve...
Homework Statement
I have to design an experiment with 3 factors. One factor has to be quantitative with at least 3 levels. One Qualitative with at least 3 levels. And the last one can be either quant/qual with at least 2 levels.
My question is in regards to coding the variables. For example...
http://sphotos-h.ak.fbcdn.net/hphotos-ak-ash4/485580_10200547582988715_854455727_n.jpg
In formula 1 it says F(1) = F(0)+F'(0)+ 0.5F''(C)
Where the heck dos the C come from? I thought they were applying taylor's formula to find an approximation of F(1), around t=0. Then c=0, right? Is it...
From the two equations given below, find ∂s/∂V (holding h constant) and ∂h/∂V (holding r constant
V = π*r^2*h, S = 2π*r*h + 2*π*r^2
Not entirely sure where to start...
Homework Statement
We have two independent, exponentially distributed random variables X and Y (with parameter a).
Z = X/(X+Y)
What is Z:s distributon function?
Homework Equations
The Attempt at a Solution
I think I need some intuition to what I'm really doing with these, I'm having a...
1. Homework Statement [/b]
this problem is on page 267 of Advanced calculus of several variables by Edwards, I just can't seem to get a handle on it:
Let aA be a contented set in the right half of the xz plane ,x>0. Define $$\hat{x}$$, the x-coordinates of the centroid of A, by...
Homework Statement
$$u_{tt} = a^2u_{xx} , 0<x< l , t>0 , $$a is constant
$$ u(x,0)=sinx , u_{t} (x,0) = cosx , 0<x< l , t>0 $$
$$ u(0,t)=2t , u(l,t)=t^2 , t>0 $$
Homework Equations
The Attempt at a Solution
I can solve the eigenvalue problem of X(x), and then solve for T(t), but...
Homework Statement
Find correlation between random variables x and y in the following:
$$P_{x,y}(x,y)=A \ xy \ e^{-(x^2)}e^{-\frac{y^2}{2}}u(x)u(y)$$
Homework Equations
The co-variance ##\sigma_{xy}=\overline{(x-\bar{x})(y-\bar{y})}## or ##\sigma_{xy}=\overline{xy}-\bar{x}\bar{y}##...
I didn't post this in the probability section cause the questions I have are more regarded to communication system engineering.
I haven't actually been able to wrap my head around these concepts mainly cause all the study material I use have these really ambiguous explanations of each...
Hi,
I would certainly appreciate it if you could please confirm the result I obtained to the following Statistics problem.
Homework Statement
A tank is supplied with fuel once a week. If the fuel (in thousands of liters) that the station sells in a week is a random variable which is...
Homework Statement
The question is: Let \phi: \mathbb{R}^n\rightarrow\mathbb{R}^n be a C^1 map and let y=\phi(x) be the change of variables. Show that dy_1\wedge...\wedge dy_n=(detD\phi(x))\cdotdx_1\wedge...\wedgedx_n.
Homework Equations
n/a
The Attempt at a Solution
Take a look at here and...
The question is: Let $\phi:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a $C^1$ map and let $y=\phi(x)$ be the change of variables. Show that d$y_1\wedge...\wedge $d$y_n$=(detD$\phi(x)$)$\cdot$d$x_1\wedge...\wedge$d$x_n$.Take a look at here and the answer given by Michael Albanese:
differential...
Hi everyone,
I have the following exercise.
Fx(x)=0, x<-1 or x>1
Fx(x)=1/2, x=[-1;1]
g(x)=x^2+1 --- this is the function of random variable
I must calculate Fy which is the sum of solutions of g(xk)=y , Fy(y)=sumFx(xk)/|g`(xk)|
g(x) is bijective on [-1;1]
y=x^2+1=> x=+sqrt(y-1) or x=-sqrt(y-1)...
Hi,
I'm trying to find a probability distribution (D) with the following property:
Given two independent stochastic variables X1 and X2 from the distribution D, I want the difference Y=X1-X2 to have a uniform distribution (one the interval [0,1], say).
I don't seem to be able to solve it...
For some time now I've been trying to figure out probably for a problem of the following form.
Say a criminal profiler is trying to determine the probability that someone is a criminal based on statistical information.
60% of people who have mustaches are criminals.
70% of people who...
Hi Everyone!
I have two normally distributed random variables. One on the x axis, the other on the y axis, like a complex normal random variable. I'm trying to find the pdf of the angle between a fixed point on the x-y plane(let's say point 1,0) and the vector formed by combining the two...
I've found many articles online that explain how to solve the Schrodinger equation for a potential dependent on x, but not for one dependent on t. A couple articles said that you could not use separation of variables to solve the Schrodinger equation with a time dependent potential, but they did...
Suppose I have a Schrodinger equation for two interacting particles located at x and y; so, something like
\left( i \frac{\partial}{\partial t} + \frac{1}{2m_x} \frac{\partial^2}{\partial x^2} + \frac{1}{2m_y} \frac{\partial^2}{\partial y^2} + V(x-y) \right) \psi(x,y,t) = 0.
Now, I want to...
Homework Statement
Consider a Hamiltonian involving two Gaussian variables, X and Y. Start from the statement that the average formed by these two variables is of the form
<e^{aX+bY}> = e^{a^2+b^2-ab}
Homework Equations
<e^{ax}> = \int_{-\infty}^{\infty} dx \frac{exp(-\beta (...
Oke this is a simple question but it has me a bit stumped.
Given a random variable X with a uniform probability distribution between [0,2].
What is the probability distribution function (pdf) of X^2 ?
Hi everyone,
I would like to know if this stament is true or not. I have two variables u,v both of them distributed as normal distribution with mean 0 and variance a^2. Is it true that the expected value of uv is a^2 ?
Thanks
Homework Statement
Let X and Y have the joint probability density function f(x,y)=k(1-y), if 0<x<y<1 and 0 elsewhere.
a)Find the value of k that makes this pdf valid.
b) Find P(X<3/4,Y>1/2)
c) Find the marginal density function of X and Y
d) Find the expected value and variance of X and...
Ok - this is a moderately tough question which I can't figure out.
So I am trying to work on a simplified model to start with.
I imagine a solid, very massive impenetrable object.
I have a tube or any long object which can exhibit some elastic behavior and also plastic behavior...
Homework Statement
The joint PDF (probability density function) ##p_{X,Y}(x,y)## of two continuous random variables by:
$$ p_{X,Y}= Axy e^{-(x^2)}e^{\frac{-y^2}{2}}u(x)u(y)$$
a) find A
b) Find ##p_X (x), \ p_{y}, \ p_{X|Y}(x|y), and \ p_{Y|X}(y|x)##
Homework Equations
The first...