## Measurement Theory – With what formalism can random variables be treated algebraically like real or complex numbers?

We all know that there is meaning when we have a variable x – for example –
$$y = e ^ x$$,

And we all know how to manipulate and calculate this algebraically. For example when

$$y_1 = e ^ {x_1}$$

and

$$y_2 = e ^ {x_2}$$

then

$$y_1y_2 = e ^ {x_1 + x_2}$$

Random variables, however, are a completely different beast. Ultimately, we all know intuitively what we mean when we draw a PDF, or more precisely the CDF, which itself is an algebraic function map $$[-infty, infty]$$ to $$[0,1]$$ as part of it defines a random variable.

But what is the actual math that makes this definition of a camper actually algebraic, so that we can happily write things like above

$$Y_1 = e ^ {X_1}$$

and

$$Y_2 = e ^ {X_2}$$

then

$$Y_1Y_2 = e ^ {X_1 + X_2}$$

from where $$X_1$$ and $$X_2$$ are random variables – and come through it ???

## real analysis – \$ f \$ is Lipschitz and \$ X \$ is zero. Show that \$ f (X) \$ is also zero.

If $$f[a,b] to mathbb {R}$$ is Lipchitz and $$X subset [a,b]$$ has zero measure show that $$f (X)$$ also has the measure zero.

What I did:

As $$X$$ has measured zero, $$forall epsilon> 0$$ There is a countable union of open intervals $$I_i$$ s.t. $$X subset cup_ {i = 1} ^ infty I_i$$ and $$sum_ {i = 1} ^ infty | I_i | < epsilon.$$ As $$f$$ is continuous $$f (I_i) = J_i$$ is an interval and $$f (X) subset cup_ {i = 1} ^ infty J_i$$, As $$f$$ is Lipschitz $$| J_i | leq k | I_i |$$, for some $$k <0.$$ Now $$sum_ {i = 1} ^ infty | J_i | and so you can take $$epsilon = frac { epsilon_1} {k}$$ break up.

Law?

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## Can an orthogonal unsymmetrical 3×3 matrix have 3 real eigenvalues?

I was wondering if a non-symmetric orthogonal matrix can have its 3 eigenvalues ​​in the real numbers.

All 3 real eigenvalues ​​of the orthogonal matrix I found are symmetric.

Can someone give me a counterexample?

## co.combinatorics – Taylor's polynomials and loss of real roots

Real rooting, logarithmic concavity and unimodality are intertwined properties. In that light, I was asked to ask the following question.

Suppose the roots of a polynomial $$p (x)$$ are all real and $$p (0)> 0$$, Fix a whole number $$k geq0$$ and look at the function $$f = frac1 {p ^ {2k + 1}}$$, I like to look at the partial sums (polynomial)
$$sum_ {j = 0} ^ {2k} frac {f ^ {(j)} (0)} {j!} x ^ j; tag1$$
from where $$f ^ {(j)}$$ Does that mean $$j$$-th derivative.

QUESTION. Is it true that the polynomial in (1) has no real roots?