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2
votes
1answer
45 views
Does $\operatorname{MSE}(\hat{\theta}) = \operatorname{Var}(\theta)+ \left(\operatorname{Bias}(\hat{\theta},\theta)\right)^2$?
We know that
$\operatorname{MSE}(\hat{\theta})=\operatorname{E}\left[(\hat{\theta}-\theta)^2\right]$ and
$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+ ...
2
votes
0answers
69 views
Why matrix representation of convolution cannot explain the convolution theorem?
A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
4
votes
2answers
77 views
Intuitive/Visual proof that $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$ [duplicate]
$$(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$$
I noticed this only because $\displaystyle \sum_{i=1}^n i = \frac{n(n+1)}{2}$ and $\displaystyle \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}$.
But the two things ...
3
votes
3answers
51 views
Prove the following is a tautology
I was trying to prove this statement is a tautology without using truth tables. Something doesn't add it here as I keep getting stuck. Take a look please!
For statements, P, Q and R prove that ...
1
vote
2answers
81 views
If $Ax=b$, for $b\ne 0$, has more than one solution, then $Ax=0$ does as well. T or F. Prove this.
I get that this is true, because there's one free variable, so no matter what the augmented matrix is, there always will be an infinite amount of solutions. Right? But how to I explain this as a ...
19
votes
0answers
249 views
Geometric interpretation for sum of fourth powers
Summing the first $n$ first powers of natural numbers:
$$\sum_{k=1}^nk=\frac12n(n+1)$$
and there is a geometric proof involving two copies of a 2D representation of $(1+2+\cdots+n)$ that form a ...
3
votes
4answers
156 views
Proof-without-words for $\bar a\times (\bar b\times\bar c)=\bar b (\bar a\cdot\bar c)-\bar c (\bar a\cdot \bar b)$ or some visual-biased explanation?
Griffiths' Introduction to Electromagnetism -book has equations called 20.10 below.
I have proved this equation d) pretty much on the first mathematics -course I had but I have not yet understood a ...
