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Jun
29
comment Cumulative distribution function of Cauchy distribution
Beware that $(1/X<x)\ne(X>1/x)$ in general.
Jun
29
comment Joint probability density for independent variables
Re (b) you suggest to compute $P(X_2<1)$, not $P(X_1>1,X_2<1)$. Re (c) you suggest to compute $P(X_1<2,X_2<2)$, not $P(X_1+X_2<2)$.
Jun
29
comment proving that $\lim_{n\to \infty}P(A_n)$ exists and $\lim_{n\to \infty}P(A_n) =P(\lim\sup A_n)$
Unfortunately, the condition that $\limsup\, [x_n - x_{n+1}] \leqslant 0$ does not imply that the sequence $(x_n)$ converges to $0$. Exercise: Find $(x_n)\subset[0,1]$ such that $x_n - x_{n+1}\to 0$ but $\limsup\,x_n=1$ while $\liminf\,x_n=0$.
Jun
29
comment What's happening at $a=-1$ in $\int x^a dx$?
$$\lim_{a\to-1}\int x^a dx=\lim_{a\to-1}\frac{x^{a+1}-1}{a+1}+C=\ln x+C$$
Jun
29
comment What to show for convergence in probability?
Voting to close. There is no question here since one cannot seriously hope this to hold for every $f:M\to M$ and every probability $P$ on $M$.
Jun
29
comment Is here $c^{-n}\in O(e^{-5n})$?
"$\frac{1}{c^n}\leqslant e^{-5n}$ for $n$ large" is indeed nonsense.
Jun
29
comment Is here $c^{-n}\in O(e^{-5n})$?
@dalastboss Still standing by your first comment?
Jun
29
revised Non linear Differential Equation
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Jun
29
comment Product Topology and Borel-$\sigma$-algebra
Then you could post a proof of this fact as an answer, to make this page useful to future readers.
Jun
29
comment What is the best answer from choices for 15:220 :: 100:?
Meaning of : and ::?
Jun
29
comment Product Topology and Borel-$\sigma$-algebra
Can you show this topology is a sigma-algebra?
Jun
29
comment Solution of $(x^2 + y^2)\ dx -2xy\ dy$ = 0
+1. Quite nice.
Jun
29
comment Solution of $(x^2 + y^2)\ dx -2xy\ dy$ = 0
Wrong minus sign at the beginning of the RHS of the first identity.
Jun
29
comment Product Topology and Borel-$\sigma$-algebra
Tautology: Every topology T which is a sigma-algebra is the Borel sigma-algebra generated by T since T is (trivially) the smallest sigma-algebra containing T.
Jun
29
revised Linear Combination of the Normal Distribution Two missing variables?
rolled back to a previous revision
Jun
29
comment Convergence of a sequence
@Bey Sorry but Markov is really unrelated to the question asked here (which is pure real analysis, by the way).
Jun
29
comment What distribution has $X^n$ if $X$ is normal distributed?
...And when $n$ is odd, $E(X^n)=0$ is direct, without computing the density.
Jun
29
revised What distribution has $X^n$ if $X$ is normal distributed?
added 10 characters in body
Jun
29
revised What is the significance of multiplying 2 Gaussian PDFs?
added 78 characters in body
Jun
29
revised What distribution has $X^n$ if $X$ is normal distributed?
added 97 characters in body; edited tags