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.

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.

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  1. R

    B Binary variables (Absolute values)

    Hello, According to https://www.fico.com/fico-xpress-optimization/docs/latest/mipform/dhtml/chap2s1.html?scroll=ssecabsval the formula for absolute values are : y = | x1 - x2| for two variables x1, x2 with 0 ≤ xi ≤ U Introduce binary variables d1, d2 to mean d1 : 1 if x1 - x2 is the positive...
  2. C

    B Question about the limits of integration in a change of variables

    Hello everyone, If I have an integral ##\int_0^r \sqrt{(r^2 - x^2)}dx## and I'm integrating across the first quadrant to get the area of the first quater of a circle. And I change variables with ##x = r\cos{\theta}## and ##dx = -{r}\sin{\theta}{d\theta}## And I form a new integral that's...
  3. docnet

    Expectation of a sum of random variables

    $$\begin{align*} E[(A+B)^2]&=E[A^2+2AB+B^2]\\ &=E[A^2]+2E[AB]+E[B^2]\\ &=2E[AB]+E[B^2]. \end{align*}$$ Can the terms ##2E[AB]## and ##E[B^2]## be simplified any more? Thanks, friends.
  4. paulb203

    B Inverse Functions: Why rewrite as y=f(x) ?

    By rewriting, for example, f(x)=2x+3, as y=2x+3, are we simply stating that something = 2x+3; and in the first case we’re calling that something f(x), and in the second case we’re calling it y? Does the y have anything to do with the y axis as in x,y coordinates axes? Or is just a randomly...
  5. cryforhelp104

    I What does a lower case r mean over variables in an equation?

    I've attached an example here.
  6. D

    Find the relation between 2 variables

    Here is the equation I obtain after simplification, I don't know if it is correct: gmc * V1 + s * C2 * Vout = [{s * (C1 + C2) * ro2 + 1} * Vout - s * C1 * ro2 * V1] * (s * rb * C2 + 1) / {ro2 * rb * (s * C2 - gm2)} I need to eliminate V1 to find the relation between Vin and Vout.
  7. D

    The Day the Earth Stood Still

    This model diagram is possible when index0/index1=index1/index2=radius2/radius1 for the model's variables. 30 degree wave into a sphere. Reminded me of something out of that old science fiction movie, The Day the Earth Stood Still...
  8. T

    A Separation of variables is possible only in 11 coordinate systems?

    I vaguely (strong word there because I can no longer remember the source, but the idea sticks in my head for 30 years now) recall reading (somewhere long forgotten) that method of separation of variables is possible in only 11 coordinate systems. I list them below: 1.Cartesian coordinates...
  9. chwala

    Conceptual question on equations of the form ##x=ay^2+by+c##

    Now i just need some clarification; we know that quadratic equations are equations of the form ##y=ax^2+bx+c## with ##a,b## and ##c## being constants and ##x## and ##y## variables. Now my question is... can we also view/look at ##x=y^2+2y+1## as quadratic equations having switched the...
  10. G

    Computing path integral with real and Grassmann variables

    The first step seems easy: computation of the $\theta$ and $\overline{\theta}$ integrals give $$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \det(\partial_j w_i(x)) \exp{\left(-\frac{1}{2}w_i(x)w_i(x)\right)}.$$ From here, I tried using that $$\det(\partial_j w_i (x)) = \det\left(\partial_j w_i...
  11. F

    I When can Ordinal Variables be treated as Interval Variables?

    Hello, Ordinal variables (see Likert scale) can be labelled using numbers and ranked by those numbers. However, the difference between category 2 and category 3 may not be exactly be the same as the difference between category 4 and 5. That said, I noticed that in social science ordinal...
  12. milkism

    Separation of variables in spherical coordinates (electrostatics)

    Problem: Solution: When I looked at an example problem, they started writing the potential in terms of the Legendre polynomials. The example problem: This is what I did: $$V_0 \alpha P_2 (\cos(\theta)) \Rightarrow \frac{\alpha 3 \cos ^2 (\theta)}{2} - \frac{\alpha}{2} \Rightarrow \frac{\alpha...
  13. WMDhamnekar

    I Expected number of random variables that must be observed

    In my opinion, answer to (a) is ## \mathbb{E} [N] = p^{-4}q^{-3} + p^{-2}q^{-1} + 2p^{-1} ## In answer to (b), XN is wrong. It should be XN=p-4q-3 - p-3 q-2- p-2 q-1 - p-1. This might be a typographical error. Is my answer to (a) correct?
  14. C

    Free variables for a matrix in REF

    For this, I am not sure what the '2nd and 5th the variables' are. Dose someone please know whether the free variables ##2, 0, 0## from the second column and ##5, 8, \pi##? Or are there only allowed to be one free variable for each column so ##2## and ##5## for the respective columns. Also...
  15. Magnetons

    PDE and the separation of variables

    using the equation ##u(x,y)=f(x)g(y)##, first, I substitute the value of ##u_{xx}## and ##u_{yy}## in the given PDE. after that solve the ODEs but I can't understand about the ##u_{t}##.In my solution, I put ##u_{t}=0## because u is only the function of x and y. Is it the right approach, to me...
  16. F

    I Handling categorical variables in R

    Hello R users, My general understanding is that, in R, nominal categorical variables (with 2 or more levels) must be first converted into factors and THEN to dummy variables (k-1 dummy variables for k levels). Is that correct? Once we accomplish categorical variable -> factor -> dummy...
  17. V9999

    I A doubt about the multiplicity of polynomials in two variables

    Let ##P(x,y)## be a multivariable polynomial equation given by $$P(x,y)=52+50x^{2}-20x(1+12y)+8y(31+61y)+(1+2y)(-120+124+488y)=0,$$ which is zero at ##q=\left(-1, -\frac{1}{2}\right)##. That is to say, $$ P(q)=P\left(-1, -\frac{1}{2}\right)=0.$$ My doubts relie on the multiplicity of this point...
  18. PainterGuy

    B Quantum entanglement and hidden variables

    Hi, I have a basic understanding of quantum physics. I was reading a Wikipedia article on hidden variables, https://en.wikipedia.org/wiki/Hidden-variable_theory . The article says the following. I was confused about the words "local" and "nonlocal" in the quote above so I checked out another...
  19. crememars

    Identifying variables from Riemann sum limits

    Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...
  20. Viona

    Integral with different variables

    I want to do this integral in the picture: where r1 and a are constants. I know I can integrate each part separately. There will be an integral with respect to r2 multiplied by integral with respect to theta2 and the last one with respect to phi2. But the term under square root confuses me. Can...
  21. B

    A How to separate variables in this PDE?

    My PDE: F,x,t + A(x)*F(x,t)*[(x+t)^(-3/2)] = 0 A(x) is a known function of x. Trying to separate F(x,t) like F(x,t) = F1(x)*F2(t)*F3(x+t). I’m getting desperate to solve, any suggestions??
  22. DrChinese

    A How Many Hidden Variables Would It Take to Model Photon Polarization?

    Suppose the EPR* concept were true as it would be applied to entangled photon polarization. (Please note: I am not saying it is.) They thought QM was incomplete, because there must exist "elements of reality" (hidden variables) that supplied the highly correlated results on entangled particle...
  23. F

    I Count and Categorical Variables...

    Hello, In the context of categorical variables, a frequency table which gives us the count (aka frequency) for each level of the categorical variable. Count is a number telling us how many times a specific level occurs. A bar-chart handles a single categorical variable (nominal or ordinal) with...
  24. G

    Calculate limit value with several variables

    Hi, I had to calculate the entropy in a task of a lattice gas and derive a formula for the pressure from it and got the following $$P=\frac{k_b T}{a_0}\Bigl[ \ln(\frac{L}{a_0}-N(n-1)-\ln(\frac{L}{a_0}-nN) \Bigr]$$ But now I am supposed to calculate the following limit $$\lim\limits_{a_0...
  25. N

    A Non-significant variables in a logistic regression model

    If some variables of a logistic regression models are non significant, should they be considered for a risk index calculation? Should the logistic model include only relevant variables? Thanks for the attention.
  26. F

    I Linear regression and random variables

    Hello, I have a question about linear regression models and correlation. My understanding is that our finite set of data ##(x,y)## represents a random sample from a much larger population. Each pair is an observation in the sample. We find, using OLS, the best fit line and its coefficients and...
  27. K

    I Canonical quantization of Ashtekar's new variables

    Is Loop quantum gravity canonical quantization of Ashtekar's new variables correct ? if not in principle is there any particular ways to canonical quantization of Ashtekar's new variables ? are there other methods to quantization of Ashtekar's new variables ?
  28. F

    I Categorical vs Numerical Variables

    Hello, I am generally clear on the distinction between numerical and nonnumerical (also called categorical or qualitative) variables but I still have some doubts in some regards. A numerical variable (continuous or discrete) has a value that derives from a measurement procedure (using a tool)...
  29. Euge

    POTW Convergence of Random Variables in L1

    Let ##\{X_n\}## be a sequence of integrable, real random variables on a probability space ##(\Omega, \mathscr{F}, \mathbb{P})## that converges in probability to an integrable random variable ##X## on ##\Omega##. Suppose ##\mathbb{E}(\sqrt{1 + X_n^2}) \to \mathbb{E}(\sqrt{1 + X^2})## as ##n\to...
  30. P

    I Change of Variables in Double Volume Integral

    In Greiner's Classical Electromagnetism book (page 126) he has a derivation equivalent to the following. $$\int_V d^3r^{'} \nabla \int_V d^3r^{''}\frac {f(\bf r^{''})}{|\bf r + \bf r^{'}- \bf r^{''}|}$$ $$ \bf z = \bf r^{''} - \bf r^{'} $$ $$\int_V d^3r^{'} \nabla \int_V d^3z \frac {f(\bf z +...
  31. A

    MSE estimation with random variables

    Hello all, I would appreciate any guidance to the following problem. I have started on parts (a) and (b), but need some help solving for the coefficients. Would I simply take the expressions involving the coefficients, take the derivative and set it equal to 0 and solve? I believe I also need...
  32. C

    I Randomly Stopped Sums vs the sum of I.I.D. Random Variables

    I've came across the two following theorems in my studies of Probability Generating Functions: Theorem 1: Suppose ##X_1, ... , X_n## are independent random variables, and let ##Y = X_1 + ... + X_n##. Then, ##G_Y(s) = \prod_{i=1}^n G_{X_i}(s)## Theorem 2: Let ##X_1, X_2, ...## be a sequence of...
  33. A

    MSE estimation with random variables

    Hello all, I am wondering if my approach is correct for the following problem on MSE estimation/linear prediction on a zero-mean random variable. My final answer would be c1 = 1, c2 = 0, and c3 = 1. If my approach is incorrect, I certainly appreciate some guidance on the problem. Thank you...
  34. Seeker220

    NLM Problem: More variables than equations

    Attempt at Solution:: ##m_2## is chosen as reference frame and FBD is drawn as shown above. We get the following equations: From ##m_1## by choosing axes along and perpendicular to acceleration of ##m_1## w.r.t ##m_2## (##a'##): $$ m_1 a' + m_1 a \cos{\theta} + m_1 g \sin{\theta} = N_1...
  35. anemone

    POTW Solve 3-Variable Equation with Real Solutions | √(z-y^2-6x-26)+x^2+6y+z-8=0

    Find the real solutions for ##\sqrt{z-y^2-6x-26}+x^2+6y+z-8=0##.
  36. A

    Break a Stick Example: Random Variables

    Hello, I would like to confirm my answers to the following random variables question. Would anyone be willing to provide feedback and see if I'm on the right track? Thank you in advance. My attempt:
  37. T

    I Quick Change of Variables Question

    Hey all, I am currently struggling with a change of variables step in my calculations. Suppose the solutions ##f_{1}(x)## and ##f_{2}(x)## of the following system of differential equations is known: Now the system I wish to solve is: Upon first glance, it seems that the association ##f_{2}(-x)...
  38. A

    Probability: pair of random variables

    Hello all, I would like to check my understanding and get some assistance with last part of the following question, please. For part (d), would I use f(x | y) = f(x, y) / f(y) ? Problem statement: My attempt at a solution, not too confident in my set-up for part (d). I drew a sketch of the...
  39. A

    Probability/Random variables question

    Hello all, I am wondering if my approach is coreect for the following probability question? I believe the joint PDF would be 1 given that the point is chosen from the unit square. To me, this question can be reduced down to finding the area of 1/4 of a circle with radius 1. Any help is appreciated!
  40. mcastillo356

    Pattern of variables with absolute value exponents

    On ##x\in{(-1,1)}##, ##x\in{\mathbb{R}}##, ##\forall{n\in{\mathbb{N}}}##, ##x^{|2n|}=O(x^{|2n+1|})## Sugestions? Any answer is wellcome! Greetings, PF
  41. SchroedingersLion

    I Integral change of variables formula confusion

    Greetings all. I just got confused by the following. Consider volume integral, for simplicity in 1D. $$ V(A) = \int_{A} dz. $$ If ##z## can be written as an invertible function of ##x##, i.e. ##z=f(x)##, we know the change of variables formula $$ V(A)=\int_{A} dz= \int_{z^{-1}(A)} |z'(x)|dx...
  42. M

    Python Calling functions without recalling all variables

    Hi PF Given the following def f1(var1, var2): var3 = var1 + var2 return var3 def f2(var1, var2, var4) var4 = 10 var5 = f1(var1, var2)*var4 return var5 it is obvious function f2 does not explicitly need variables var1 and var2. However, it needs the result of f1, which...
  43. WMDhamnekar

    MHB How to use change of variables technique here?

    Find the volume V inside both the sphere $x^2 + y^2 + z^2 =1$ and cone $z = \sqrt{x^2 + y^2}$ My attempt: I graphed the cone inside the sphere as follows. But I don't understand how to use the change of variables technique here to find the required volume. My answer without using integrals is...
  44. M

    Probability that 𝑌>3𝑋 where 𝑋,𝑌 are 𝑁(0,1) random variables

    After plotting the above (not shown) I believe one way (the hard way) to solve this problem is to compute the following integral where ##f(x) = e^{-x^2/2}/\sqrt{2\pi}##: $$\frac{\int_0^\infty \int_{3X}^\infty f(X)f(Y)\, dydx + \int_{-\infty}^0 \int_0^\infty f(X)f(Y)\...
  45. WMDhamnekar

    Change of variables in multiple integrals

    Summary: Find the volume V of the solid inside both ## x^2 + y^2 + z^2 =4## and ## x^2 +y^2 =1## My attempt to answer this question: given ## x^2 + y^2 +z^2 =4; x^2 + y^2 =1 \therefore z^2 =3 \Rightarrow z=\sqrt{3}## ## \displaystyle\iiint\limits_R 1dV =...
  46. O

    MHB Partially Decoupled System with 3 variables

    Hello! I have the following initial value problem: \[ x' = x + 2y + 3z \] \[ y' = 4y + 5z \] \[ z' = 6z \] All I'm looking to do is find the general solution to this system, and as long as I'm doing this correctly I have these answers: \[ y(t) = K_2e^{4t} + \tfrac{5K_1}{2}e^{6t} \] \[...
  47. K

    I Alternative ways to quantize Ashtekar variables

    it's come to my attention that there are arguments that loop quantum gravity is simply wrong, and specifically in the way it method of quantization of Ashtekar variables. So what are more promising ways to canonically and nonperturbatively quantize Ashtekar variables Ashtekar variables are...
  48. LCSphysicist

    Finding the generator function for 4 variables

    Prove directly that the transformation $$Q_{1} = q_{1}, P_{1} = p_{1} − 2p_{2}$$ $$Q_{2} = p_{2}, P_{2} = −2q_{1} − q_{2}$$ is canonical and find a generating functionSo the first part is easy and can be skipped here. I have some difficults regarding the second part, namely, the one that ask for...