Nataly Best

2022-01-22

A quick question; is it possible to say in a way analogous to the single variable case that a multivariable function is "asymptotically equivalent" to a second multivariable function? For example, consider the function of $n}_{1},{n}_{2}\in \mathbb{R$ given by

$\text{Var}\left(\hat{\mu}\right)=\frac{{\sigma}^{2}({n}_{1}+2{n}_{2})}{{({n}_{1}+{n}_{2})}^{2}}.$

where$\sigma}^{2$ is a constant.

Can we say that$\text{Var}\left(\hat{\mu}\right)\approx \frac{1}{{n}_{1}+{n}_{2}}$ and then conclude that $\text{Var}\left(\hat{\mu}\right)\to 0$ as $n}_{1}\to \mathrm{\infty$ and $n}_{2}\to \mathrm{\infty$ ?

$\underset{(x,y)\to (\mathrm{\infty},\mathrm{\infty})}{lim}\frac{x+2y}{{(x+y)}^{2}}$

does not exist. Am I wrong to think of$\text{Var}\left(\hat{\mu}\right)$ as a function of two variables?

where

Can we say that

does not exist. Am I wrong to think of

utgyrnr0

Beginner2022-01-23Added 11 answers

The two previous answers are perfectly right, thus I just would like it emphasize the statistical perspective. Note that when you are talking about a variance of an estimator, n1 and n2 are sample sizes, hence they are (strictly) positive integers, i.e., $n}_{1},{n}_{2}\in \mathbb{N$ . As such, when you take the limit, you cannot consider any possible route (as Wolfram does) in $\mathbb{R}}^{2$ . The extreme cases in this variance are $n}_{1}>>{n}_{2$ or $n}_{2}>>{n}_{1$ , that dont

Karli Kaiser

Beginner2022-01-24Added 9 answers

You can conclude $Var\left(\mu \right)\to 0$ as $n}_{1}\to \mathrm{\infty$ and $n}_{2}\to \mathrm{\infty$ , but you can't say $Var\left(\mu \right)\approx \frac{1}{{n}_{1}+{n}_{2}}$ because when $n}_{1}>>{n}_{2$ it is approximatly $\frac{1}{{n}_{1}+{n}_{2}}$ but when $n}_{1}<<{n}_{2$ it is near $\frac{2}{{n}_{1}+{n}_{2}}$

In a regression analysis, the variable that is being predicted is the "dependent variable."

?

a. Intervening variable

b. Dependent variable

c. None

d. Independent variableWhat is ${R}^{*}$ in math?

Repeated addition is called ?

A)Subtraction

B)Multiplication

C)DivisionMultiplicative inverse of 1/7 is _?

Does the series converge or diverge this $\sum n!/{n}^{n}$

Use Lagrange multipliers to find the point on a surface that is closest to a plane.

Find the point on $z=1-2{x}^{2}-{y}^{2}$ closest to $2x+3y+z=12$ using Lagrange multipliers.

I recognize $z+2{x}^{2}+{y}^{2}=1$ as my constraint but am unable to recognize the distance squared I am trying to minimize in terms of 3 variables. May someone help please.Just find the curve of intersection between ${x}^{2}+{y}^{2}+{z}^{2}=1$ and $x+y+z=0$

Which equation illustrates the identity property of multiplication? A$(a+\mathrm{bi})\times c=(\mathrm{ac}+\mathrm{bci})$ B$(a+\mathrm{bi})\times 0=0$ C$(a+\mathrm{bi})\times (c+\mathrm{di})=(c+\mathrm{di})\times (a+\mathrm{bi})$ D$(a+\mathrm{bi})\times 1=(a+\mathrm{bi})$

The significance of partial derivative notation

If some function like $f$ depends on just one variable like $x$, we denote its derivative with respect to the variable by:

$\frac{\mathrm{d}}{\mathrm{d}x}f(x)$

Now if the function happens to depend on $n$ variables we denote its derivative with respect to the $i$th variable by:

$\frac{\mathrm{\partial}}{\mathrm{\partial}{x}_{i}}f({x}_{1},\cdots ,{x}_{i},\cdots ,{x}_{n})$

Now my question is what is the significance of this notation? I mean what will be wrong if we show "Partial derivative" of $f$ with respect to ${x}_{i}$ like this? :

$\frac{\mathrm{d}}{\mathrm{d}{x}_{i}}f({x}_{1},\cdots ,{x}_{i},\cdots ,{x}_{n})$

Does the symbol $\mathrm{\partial}$ have a significant meaning?The function $f(x,y,z)$ is a differentiable function at $(0,0,0)$ such that ${f}_{y}(0,0,0)={f}_{x}(0,0,0)=0$ and $f({t}^{2},2{t}^{2},3{t}^{2})=4{t}^{2}$ for every $t>0$. Define $u=(6/11,2/11,9/11)$, with the given about. Is it possible to calculate ${f}_{u}(1,2,3)$ or ${f}_{u}(0,0,0)$, or ${f}_{z}(0,0,0)$?

Given topological spaces ${X}_{1},{X}_{2},\dots ,{X}_{n},Y$, consider a multivariable function $f:\prod _{i=1}^{n}{X}_{i}\to Y$ such that for any $({x}_{1},{x}_{2},\dots ,{x}_{n})\in \prod _{i=1}^{n}{X}_{i}$, the functions in the family $\{x\mapsto f({x}_{1},\dots ,{x}_{i-1},x,{x}_{i+1},\dots ,{x}_{n}){\}}_{i=1}^{n}$ are all continuous. Must $f$ itself be continuous?

Let $x$ be an independent variable. Does the differential dx depend on $x$?(from the definition of differential for variables & multivariable functions)

Let $f:M(n,\mathbb{R})\to M(n,\mathbb{R})$ and let $f(A)=A{A}^{t}$. Then find derivative of $f$, denoted by $df$ .

So, Derivative of $f(df)$ if exists, will satisfy $limH\to 0\frac{||f(A+H)-f(A)-df(H)||}{||H||}=0$.if $F(x,y)$ and $y=f(x)$,

$\frac{dy}{dx}=-\frac{\frac{\mathrm{\partial}}{\mathrm{\partial}x}\left(F\right)}{\frac{\mathrm{\partial}}{\mathrm{\partial}y}\left(F\right)}$

1) $F(x,y)$ 𝑎𝑛𝑑 $y=f(x)$ so his means that the function $F$ is a function of one variable which is $x$

2) while we were computing 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑠 we treated $y$ and $x$ as two independent variables although that $y$ changes as $x$ changes but while doing the 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑠 w.r.t $x$ we treated $y$ and $x$ as two independent varaibles and considered $y$ as a constantLet $f:{\mathbb{R}}^{2}\to \mathbb{R}$ be defined as

$f(x,y)=\{\begin{array}{ll}({x}^{2}+{y}^{2})\mathrm{cos}\frac{1}{\sqrt{{x}^{2}+{y}^{2}}},& \text{for}(x,y)\ne (0,0)\\ 0,& \text{for}(x,y)=(0,0)\end{array}$

then check whether its differentiable and also whether its partial derivatives ie ${f}_{x},{f}_{y}$ are continuous at $(0,0)$. I dont know how to check the differentiability of a multivariable function as I am just beginning to learn it. For continuity of partial derivative I just checked for ${f}_{x}$ as function is symmetric in $y$ and $x$. So ${f}_{x}$ turns out to be

${f}_{x}(x,y)=2x\mathrm{cos}\left(\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}\right)+\frac{x}{\sqrt{{x}^{2}+{y}^{2}}}\mathrm{sin}\left(\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}\right)$

which is definitely not $0$ as $(x,y)\to (0,0)$. Same can be stated for ${f}_{y}$. But how to proceed with the first part?