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1d
comment Equivalent definition of singular random variable
This relies on the fact that every probability measure on the Borel sigma-algebra of the real line can be decomposed uniquely as the sum of three measures, the first one discrete, the second one absolutely continuous, and the third one singular. Most books on measure theory prove this, which one(s) are you using?
1d
comment Is the expected value of a monotone function on a uniformly distributed random variable monotone?
No wonder you could not follow, since you seem to have misread it big time... Anyway, here is another (huge) hint: If $X$ is uniform on $[a,b]$ and $a\leqslant c$ and $b\leqslant d$, can you build some $Y$ uniform on $[c,d]$ such that $Y$ is a (very) simple function of $X$ and $X\leqslant Y$ almost surely?
1d
comment Is the expected value of a monotone function on a uniformly distributed random variable monotone?
The statement that $X_n\geqslant X_{n-1}$ almost surely does not always hold (and is not in my comment). // The notation $\int X_ndX_n$ is meaningless (and horrible). // Of course $E(X)\geqslant E(Y)$ is in no way equivalent to $X\geqslant Y$ almost surely. // Sorry but your last comment seems to have been composed randomly, why is that so?
1d
comment Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$
@RonGordon But now it did get a critical comment, right? (Having said that, I fully agree with your remark about the upvotes there.)
1d
comment Is the expected value of a monotone function on a uniformly distributed random variable monotone?
Re a proof of the inequality on the right, please read my previous comment.
1d
revised Is the expected value of a monotone function on a uniformly distributed random variable monotone?
edited tags
1d
comment Is the expected value of a monotone function on a uniformly distributed random variable monotone?
I fail to understand the meaning of the implication. Both sides of the implication hold because $f$ is nondecreasing and because there exists some $X'_n$ and $X'_{n-1}$ distributed like $X_n$ and $X_{n-1}$ respectively and such that $X'_n\geqslant X'_{n-1}$ almost surely.
2d
comment Deriving sample size using Hoeffding's Inequality
Logarithm with base $e$ is often denoted $\log$ by mathematicians.
2d
comment Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$
Uniform convergence is useless on unbounded intervals. One way or another, some domination condition must be checked.
2d
comment Evaluating limits with square roots
Three problem-statement-questions in one hour. How many more in the buffer?
2d
comment Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$
The generalization holds under some further conditions, these conditions hold for gaussian integrals, but all this needs to be made much clearer in your post.
2d
comment Given the sequence of functions, $f_1(x):=\sin(x)$ and $f_{n+1}(x):=\sin(f_n(x))$, why $|f_n(x)|\leq f_n(1)$?
You might mention more clearly (since the question is misleading on this point) that the statement the OP is interested in holds for $x$ in $[-1,1]$, not for every $x$ real.
2d
comment Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$
So, for example, you did read my comment suggesting to consider the derivative with respect to beta, you pondered it in depth, you marvelled at the simplicity of the argument (like me, when I was first exposed to it), but the idea to differentiate with respect to alpha did not arise? It is a pity that none of these thoughts appear in your questions.
2d
comment Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$
You are applying to integrals on an unbounded interval, results valid for intervals on a compact interval. This cannot work.
2d
comment Find the general solution of the ODE $xy′′ − y′ + 4x^3y = 0$
Quote: "I dont need a full solution beacuse that would take hours but maybe just the final answer?"
2d
comment Convert $e^z$ to Cartesian form (complex numbers)
Isn't all this already explained in the question?
2d
comment Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$
Continuity is not the argument that allows to differentiate under the integral sign. Please reread your lecture notes or find the theorem on WP.
2d
comment Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$
Subtle? No, rather, already explained to you à propos your previous question... Do you read the comments/answers on your own questions?
2d
comment Why is the sequence $u_N = \inf\{s_n : n \gt N\}$ increasing?
Because $u_N=\inf\{u_{N+1},s_{N+1}\}$ hence $u_N\leqslant u_{N+1}$.
2d
comment Bounding a strange function
@amathnerd For no reason? How do you know? In the present case, this is rather ironic, since you wrote yourself a perfectly valid reason to downvote: the post does not answer the question (quote: "how could this help"). Additionally, the caption "Hint" is misleading. Finally, I note that you did not see fit to upvote the answer you accepted... How is that? All in all, it seems you might want to seriously rethink your approach to votes on the site...