## An estimator for Inverse of variance

I want to find some function $$f$$ such that
$$mathbb{E}left( f(X) right) = frac{1}{sigma^2},$$
where $$X sim mathcal{N}(0,sigma^2)$$.
Obviously, $$f$$ should not depend on $$sigma$$.

Notes

For a normal random variable, $$mathbb{E}(1/X^2)$$ does not exist.

## inverse number that is a prime mod \$p\$

It is possible to find the inverse of $$1/x equiv q mod p$$, with $$p$$ a prime, where q is also a prime?

## inverse – establish matrix and inversion of a uppertriangularized matrix

Trying to create an uppertriangularized matrix with Poisson Distribution, find its inverse and multiply the inverse by a vector, i.e.

``````mu=5;
lambda=158;
Daily = {a 158 element vector};
ME = UpperTriangularize(Table((mu^(lambda + 1 - j) Exp(-mu))/(lambda + 1 - j)!,
``````

{i, lambda}, {j, lambda}));
Actual = Inverse(ME).Daily;
cumulative = FoldList(Plus, 0, Actual)

the notebook would not execute and displays a redline on the right edge of the screen and a statement below the code saying

ln(2) MatrixForm(ME(tt_, mm_) := UpperTriangularize(Table((mm^(tt + 1 – j) Exp(-mm))/(tt + 1 – j)!, {i, lambda}, {j, lambda})))

Out(2) MatrixForm=
Null

## Product of Moore Penrose Inverse and Matrix

Say, we had m x n matrix B. B+ is the Moore Penrose Inverse (pseudoinverse) of the matrix. Would B+B (product of pseudoinverse of B and B) be a projection matrix? How would we prove this?

## calculus – Why the derivative of inverse secant has an absolute value?

$$y=operatorname{arcsec}(x)$$ can be defined in two ways. The first restricts the domain to $$(0,pi), xneqfrac{pi}{2}$$. In this case, the derivative is $$frac{dy}{dx}=frac{1}{|x|sqrt{x^2-1}}$$ And when the domain function is defined as $$(0,frac{pi}{2})bigcup(pi,frac{3pi}{2})$$ the derivative is $$frac{dy}{dx}=frac{1}{xsqrt{x^2-1}}$$ (It’s just different in the absolute value). But why?

In my attempt to prove it, I’m using the theorem for derivate an inverse function, which is $$frac{d}{dx}(f^{-1}(x))=frac{1}{f'(f^{-1}(x))}$$ So, we have $$frac{d}{dx}(operatorname{arcsec}(x))=frac{1}{sec (operatorname{arcsec}(x))cdottan (operatorname{arcsec}(x))}$$ At this point, I think is safe to say that $$sec (operatorname{arcsec}(x))=x$$ because this identity works for $$|x|geq1$$, and that’s a the domain of $$y$$.

For the $$tan$$, I’m using the right triangle diagram

Right Triangle Diagram

Where the hypotenuse is $$x$$, the adjacent side is 1 (so $$sec y=x$$) and the opposite side is $$sqrt{x^2-1}$$. The $$tan y$$ results in $$sqrt{x^2-1}/1=sqrt{x^2-1}$$. Substituting we get $$frac{d}{dx}=frac{1}{xsqrt{x^2-1}}$$ But I don’t understand where does the absolute value come from.

Reading other questions and watching some videos, I realize that the absolute value comes from the identity $$tan^2x+1=sec^2x$$, but as you can see, this prove doesn’t use that identity.

## camera – Why inverse the rigid transformation in perspective projection formulation?

``````    cv::Mat Rc = Rz(cam_ori(i).z) * Ry(cam_ori(i).y) * Rx(cam_ori(i).x);
cv::Mat tc(cam_pos(i));
cv::Mat Rt;
cv::hconcat(Rc.t(), -Rc.t() * tc, Rt);
cv::Mat P = K * Rt;
cv::Mat x = P * X;
``````

cv::hconcat() is horizontally concatenating the first two input matrices and store the output matrix in the third argument Rt, which is the right transformation matrix in homogeneous coordinates.

I think the X coordinates are recorded in terms of world coordinates. So, why the author implements the projection formulation with the rigid transformation being inversed?

The source code is in https://github.com/sunglok/3dv_tutorial/blob/834b1fe39583d71bb57f4a93c015a2494a78c1cc/src/image_formation.cpp#L35

## real analysis – The existence of inverse function of “rotated” \$f(x)=x^{3} sin{frac{1}{x}}\$ on the neighborhood of x=0?

I’m a bit confused about the existence of inverse function of a function. Let’s consider the “rotated” version of $$f(x)=x^{3} sin{frac{1}{x}}$$, say, rotate it counterclockwise around the origin through a small enough angle $$theta$$, to yield a wave that straddles a line $$y=kx$$, where $$k=tan{theta}$$.

Then we see since the rotation angle is very small, any straight line $$y=c$$ must intercept the figure of the function at least 2 points(when c is small enough), thus the inverse function on any neighborhood of x=0 cannot exist.

But by inverse function theorem, $$f$$ is continuously differentiable, $$f^{prime}=k>0$$, and $$f(0)=0$$, it should have inverse function on a neighborhood of x=0.

Why?

## I need to find the inverse function of f(x)=x³+x

Normally we would isolate x in terms of y in a simple equation like f(x)=4x+3 then we would write it as y=4x+3 and then write x in terms of y as y-3/4=x and so we get the inverse if it is possible. (Note): The example was only shown as a reference.

## set theory – Why are quotient sets (types) called quotients — are they the inverse of some product?

There seems to be a beautiful relation between natural numbers and sets (and types),
as in the size of a discriminate union, cartesian product, and function type,
is described by the sum, product, exponential of the sizes of the components. (As I learned from type theory). This also makes it easy to see why the symbols + and x are used for discriminate union and cartesian product (sum type and product type).

$$forall A, B, C : text{sets} \ A + B = C ~ implies ~ |A| + |B| = |C|\ A times B = C ~ implies ~ |A| times |B| = |C|\ A → B = C ~ implies ~~~~~~~~ |B|^{|A|} = |C|$$

However, why are quotient sets (and quotient types) called quotients and use the symbol $$/$$?

That does not seem to make sense to me. At the very least, to deserve the name quotient, I would expect them to somehow be the inverse of some product. I first thought they should be the inverse of the cartesian product, I tried to google this, but I cannot find anything. Is there some relation between quotient (sets) and (cartesian) products, that I am missing?

## pr.probability – In the distribution and the expectation of the inverse se Gaussien Variable

let $$H$$ be $$(N, M)$$ Gaussienne independent elements matrix where $$D$$ represent the eigenvalue matrix of the SVD decomposition of $$H$$: $$(U,D,V)=svd(H)$$ my question is:

1) What is the distribution of the inverse of the nonnull element of $$D$$
begin{align} f = frac{1}{{1 + sumlimits_{i = 1}^{Min(N,M)} {frac{1}{{D(i,i)}}} }} end{align}
2) the expectation of $$f$$?