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2d
comment Find the density of a ratio of random variables
Thus making impossible to bring said assistance in a constructive way. Back to my first remark: how did you compute the density 1/15 for every z>0?
2d
comment Find the density of a ratio of random variables
This is not necessary at all to show the steps of your reasoning.
2d
comment Random walk of a bishop
I don't see why you do not see why your question got put on hold since, as you say, it has no context.
2d
comment Find the density of a ratio of random variables
"Don't know how to show working when it's integrals." Why is that?
2d
comment Determine if these are Characteristic functions
@avid19 ?? Yes it is, and the value is zero, even.
2d
comment Find the density of a ratio of random variables
The density of Y/X is 1/15 only on z<10. And E(Y/X)=10, not 15/2. You might want to show the details of what you did.
2d
comment Finding a Lyapunov function for a given system
Sorry but, from the phase diagrams, I cannot determine whether trajectories cycle or spiral outwardly or spiral inwardly (and I would be cautious about approximation errors in simulations, if I were you).
2d
comment Is $\overline{z} $ independent of $z $?
?? We do not arrive at the same formula - perhaps you could explain more precisely what you are thinking about. (Unrelated: Adding - just after @user seems to inactivate the notification of the comment.)
2d
comment independence and characteristic functions
Flawed logical circle: this shows that independence implies the property, not the reverse.
2d
comment Laplace vs. non-Laplace Solution of ODE
Use uniqueness of solutions twice, thus $y(t)=0$ for $t<1$ and, for $t\geqslant1$, $$y(t)=1-e^{1-t}.$$
2d
comment There exist a set $X$ such that the number of function $y: x\to \{1,2,3\}$ is $1000$.
@CliveNewstead $3^6$ and $3^7$? :-)
2d
comment Finding a Lyapunov function for a given system
Simulated phase diagrams do not seem conclusive enough to decide whether (0,0) is stable. Why do you think it is?
2d
comment Find a Liapunov function to show asymptotically stable
$$V(x,y)=x^2+xy+y^2\qquad D=\{(x,y)\mid x^2+y^2\leqslant\tfrac12\}$$
2d
comment Is $\overline{z} $ independent of $z $?
"This seems to imply that $\overline {z}$ does not depend on $z $." Why do you think so?
2d
comment Why is $\sqrt{X}\times\sqrt{X}=X$?
"The square root of a real number $x\ge0$ is another" NONNEGATIVE "real number $a$ such that $a^2=x.$"
2d
comment Existence of divergent series $\sum_{n=1}^ \infty a_n$ of real numbers whose partial sums are bounded and $\lim (na_n)=0$
@Chou ?? What are you referring to?
2d
revised How to plot a phase portrait for system of differential equations in mathematica or R?
added 224 characters in body
2d
comment how to show that all solutions tend to zero?
Phase diagrams provide this in a whiff: for every $t\geqslant-\log\varepsilon$, $e^{-t}$ is in $(0,\varepsilon)$ and $y'=-2y(1+\frac12y^2)+e^{-t}$ hence $y'<0$ for every $y>\frac12\varepsilon$ and $y'>0$ for every $y<0$ thus every solution $y$ ends up in the interval $(0,\frac12\varepsilon)$, QED. Exercise: Write down a wide-ranging generalization of the result dealing with solutions of differential equations $y'+G(y)=H(t)$.
2d
answered How to plot a phase portrait for system of differential equations in mathematica or R?
2d
comment Convergence of probability for $t$-distribution
I fail to see how $x<-1$ is a problem since anyway, if $x<-1$ then, for every $n$, the event $[T_n^2+2T_n\leqslant x]$ is the empty set. In particular, the value of $\lim\limits_{n\to\infty}P(T_n^2+2T_n\leqslant x)$ is rather clear.