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Different rules for calculating the variance of a sum
So apparantly there are at least three rules for calculating the variance of a sum; $\mathrm{Var}\left(\sum_{i=1}^nX\right)$. When are each rule valid? Ar...
Apr 22, 2026
Archive: News
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Proof of elementary row operations for matrices?
I'm taking a Linear Algebra course, and we just started talking about matrices. So we were introduced to the elementary row operations for matrices w...
Apr 22, 2026
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Isoperimetric problem with constraint
Solve the isoperimentric problem to minimize $$\int_0^1 y(x)\,{\rm d}x$$ subject to the constraint $$\int_0^1 \sqrt{1 + y'(x)^2}\,{\rm d}x = \frac{\p...
Apr 22, 2026
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Finding a Correlation between Bernoulli Variables?
Let X and Y be Bernoulli random variables. We don't assume independence or identical distribution, but we do assume that all 4 of the following proba...
Apr 22, 2026
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Determine all convex polyhedra with $6$ faces
I want to determine all convex polyhedra with 6 faces (not necessarily regular). Based on the Euler characteristic, $v-e+f=2$, we know that $v-e+6=2$, or ...
Apr 22, 2026
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Expected number of steps for reaching $K$ in a random walk
Assuming steps are $+1/-1$ with a $50/50$ probability. What is the expected step count for reaching $10, 100$ or $K$?
Apr 22, 2026
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Why is the composition of a function and its inverse commutative?
I've read that the composition of functions is not commutative but in this case it works. So how can we proof that the composition of a function and ...
Apr 22, 2026
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Derive parametric equations for sphere
How do you derive the parametric equations for a sphere? \begin{align} x & = r \cos(\theta)\sin(\varphi), \\ y & = r \sin(\theta)\sin(\varphi), \\...
Apr 22, 2026
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SSS Congruence Theorem
I was wondering whether there is a proof of SSS Congruence Theorem (and also whether there is one for SAS and ASA Congruence Theorem). In my textbook, the...
Apr 22, 2026
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Defining addition of vectors of different dimensions
While doing real data analysis I came up with a problem. I have given lots of efforts to solve it and could not succeed. Here is the problem: Suppose, we ...
Apr 22, 2026
