Intereting Posts

Find all integer solutions of $1+x+x^2+x^3=y^2$
What is a homomorphism and what does “structure preserving” mean?
Conceptualizing Inclusion Map from Figure Eight to Torus
Galois theory: splitting field of cubic as a vector space
Calculating volume of convex polytopes generated by inequalities
Proof of Bolzano's Theorem
Find the asymptotic tight bound for $T(n) = 4T(n/2) + n^{2}\log n$
Factorial of a large number and Stirling approximation
Poisson Integral is equal to 1
How to find a confidence interval for a Maximum Likelihood Estimate
Prove that $\dfrac{|x+y|}{1+|x+y|}\leq\dfrac{|x|}{1+|x|}+\dfrac{|y|}{1+|y|}$ for any $x,y$
When is the derived category abelian?
Can contractible subspace be ignored/collapsed when computing $\pi_n$ or $H_n$?
Why do we use both sets and predicates?
Two definitions of Lebesgue integration

I recently know that following results.

suppose that $x_1, x_2, x_3$ are independent real Gaussian random variables with $\mathcal{N}(0, 1)$. Then

$$

\frac{x_1 + x_2 x_3}{\sqrt{1+x_3^2}} \sim \mathcal{N}(0, 1)

$$

- Summing (0,1) uniform random variables up to 1
- Why does this not seem to be random?
- Mathematicians shocked(?) to find pattern in prime numbers
- How can we generate pairs of correlated random numbers?
- If we would have a perfect random decimal number generator, what would the chances be for the occurence of the numbers?
- Nonlinear transform of two random variables for Gaussianity

We can prove this result by direct computing. But I am wondering if there is a simpler way. Also, since this result is interesting. I am wondering if there is any generalization

Thanks

- “Random” generation of rotation matrices
- Show that $C_1= [\frac{k}{2^n},\frac{k+1}{2^n})$ generates the Borel σ-algebra on R.
- $X = E(Y | \sigma(X)) $ and $Y = E(X | \sigma(Y))$
- Looking for strictly increasing integer sequences whose gaps between consecutive elements are “pseudorandom”
- Does an infinite random sequence contain all finite sequences?
- Are squares of independent random variables independent?
- Is the product of two Gaussian random variables also a Gaussian?
- Question on the 'Hat check' problem
- How to find distribution of order statistic
- Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

The point is that the conditional distribution of your random variable given $x_3$ is always ${\cal N}(0,1)$. One generalization is this. Suppose

$X_1, \ldots, X_n$ are independent ${\cal N}(0,1)$ random variables, and ${\bf Y} = (Y_1, \ldots, Y_n)$ is a vector-valued random variable independent of $X_1, \ldots, X_n$ and supported on the sphere $Y_1^2 + \ldots + Y_n^2 = 1$. Then

${\bf X} \cdot {\bf Y} = X_1 Y_1 + \ldots + X_n Y_n \sim {\cal N}(0,1)$.

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- Is there a common notion of $\mathbb{R}^n$, for non-integer $n$?
- Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
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- Find the point in a triangle minimizing the sum of distances to the vertices
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- Find $\lim_{x\to0}\frac{\sin5x}{\sin4x}$ using $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$.
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- Longest increasing subsequence part II
- Is there any intuition behind why the derivative of $\cos(x)$ is equal to $-\sin(x)$ or just something to memorize?
- Proving that $ 2 $ is the only real solution of $ 3^x+4^x=5^x $