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 $f$?