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21h
comment Consider 2 Stocks. If Stock 1 sells \$10(0.8) or sells \$20(0.9). If Stock 2 sells \$10(0.9) or \$25(0.8). Which stock sells for higher price?
You should try instead to solve the following: One given stock always sells for x or y. If it is selling for x today, there are p chances that it will sell for y tomorrow. If it is selling for y today, there are q chances that it will sell for x tomorrow. On the average, which price this stock will sell for? The answer will depend on (x,y,p,q). Then apply this result to stock 1 and to stock 2 and compare the average prices each stock sells for.
21h
comment What's the difference between a random variable and a measurable function?
These are the same. Searching for a difference, one could note that every measurable function on a measurable space becomes a random variable as soon as one fixes a probability measure on the measurable space to give it the structure of a probability space. Thus, random variables would be measurable functions defined on a probability space, or, to summarize: random variable = measurable function + probability measure.
21h
comment Identifying markov chains and the markov property
The fact that you think this is all very well but the site asks to explain reasons why you have the proposition of answer you have.
21h
comment Marginal distribution for two continous random variables
One option, explained several times on the site, is to write rigorously the PDF of $(X,Y)$ from the start, here, $$f_{X,Y}(x,y)=a\mathbf 1_{x^2<y<-x}\mathbf 1_{-1<x<0},$$ and to proceed in an automatized way from there. For example, $$1=\iint_{\mathbb R^2} f_{X,Y}(x,y)dxdy=a\iint_{\mathbb R^2} \mathbf 1_{x^2<y<-x}\mathbf 1_{-1<x<0}dxdy=a\int_{-1}^0\int_{x^2}^{-x}dydx=\ldots$$ yields $a$ and $$f_Y(y)=\int_\mathbb R f_{X,Y}(x,y)dx,$$ valid with no restriction on $(x,y)$, yields $f_Y$.
22h
comment Marginal distribution for two continous random variables
This does not define $f_{XY}$. Please revise.
22h
comment Prove $\ell^5$ is contained in $\ell^6$.
5 users find this answer useful. I have no idea how it can lead to a solution.
23h
comment Markov factorization of the density of an AR(1) process
This seems to be a direct consequence of Bayes formula, applied recursively to $(X_k)_{1\leqslant k\leqslant n}$. Alternatively, one can note that $(X_k)_{1\leqslant k\leqslant n}$ is a linear transform of $X_1$ and $(Z_k)_{2\leqslant k\leqslant n}$, all independent, and apply a (very simple) form of change of variable formula.
1d
comment Does the law of large numbers pin down the distribution of an infinite sample?
The answer is obviously yes and, since every sensible definition of the limiting frequencies of a sample involves a countable sample, considerations of alephs are quite offtopic here. To prove the result, indeed the SLLN suffices (see @M.Wind's comment, replacing the cautionary "should" and "presumably" by affirmatives). For a quantification of the size of the sample needed for a given threshold of the error, look for "Glivenko-Cantelli".
1d
comment How to describe behavior of population system, given by system of ODEs?
I am afraid you might have missed that the (excellent) answer you accepted mentions (rather briefly) that there exists a systematic approach to answer this kind of question, not based on visualization tools (that are also quite useful) but on actual mathematics. Are you aware of this fact? I am asking you this because of your comment mentioning "the convergence of $f_1(x,y)$", quite offtopic, which seems to indicate otherwise.
1d
comment I need help to solve this complex question
OP: Are you currently in the process of trying to cover your tracks by defacing every question you asked on the site? You might want to know that such behaviour is very much frowned upon.
1d
revised Gauss Method to show
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1d
comment Gauss Method to show
Please do not deface your question, even if it is currently closed as off-topic.
1d
comment Find closed form formula
Please do not deface your question.
1d
revised Find closed form formula
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1d
comment Show the closed form of the sum $\sum_{i=0}^{n-1} i x^i$
Please do not deface your question, even if it is currently marked as duplicate.
1d
revised Show the closed form of the sum $\sum_{i=0}^{n-1} i x^i$
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1d
comment I need help to solve this complex question
Please do not deface your question, even if it is marked as duplicate.
1d
revised I need help to solve this complex question
rolled back to a previous revision
1d
revised Tricky Cardinality Question/Riddle
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1d
comment Tricky Cardinality Question/Riddle
Please do not deface your question.