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Jun
28
comment $X_1$, $X_2$ i.i.d., prove that $E(X_1\mid X_1+X_2) = E(X_2\mid X_1+X_2)$
Hint: $E(h(U+V)U)$ depends only on the joint distribution of $(U,V)$. Apply this to $(U,V)=(X_1,X_2)$ and $(U,V)=(X_2,X_1)$.
Jun
28
comment Jacobian Transformation p.d.f
But now your f_V is not a density.
Jun
28
comment Why is there root of -1?
"They" (were Italians of the XVIth century and they) wanted to find the roots of third degree polynomials. It happened that they found themselves literally forced to use imaginary numbers, first as a trick providing rather mysteriously the solutions, and then, after a while, progressively, as a mathematical object worth investigating. Heroes of the time: Tartaglia, Cardano, Bombelli.
Jun
28
comment Jacobian Transformation p.d.f
The density of V cannot depend on some parameter u. Also, see my comment to the question.
Jun
28
comment Jacobian Transformation p.d.f
The support of (U,V) is not the set of (u,v) such that u>v>0 since U>2V almost surely.
Jun
28
comment How can I find the density of $E[X\mid Y]$ when $(X,Y)$ is gaussian
Sorry, I do not check handwritten proofs presented in images.
Jun
28
comment How can I find the density of $E[X\mid Y]$ when $(X,Y)$ is gaussian
The theory says that $E(X\mid Y)=aY+b$ and that $(a,b)$ is entirely determined by the pair of identities $$E(X)=E(E(X\mid Y))=aE(Y)+b\quad E(XY)=E(E(X\mid Y)Y)=aE(Y^2)+bE(Y)$$ Thus, $E(X\mid Y)$ is normal with mean $aE(Y)+b$ and variance $a^2\sigma^2_Y$. Can you finish?
Jun
28
comment Is $\tau$ a topology on $\mathbb{R}^2$? where the elements of $\tau$ are $\emptyset$ and the complements of finite sets of lines and points
What are the axioms defining a topology, already?
Jun
28
comment Markov chain state reached earlier than other state
If really the question is to know the chance, starting from state 3, to reach state 1 earlier than state 4, one may want to note that p(3,1)=p(3,4)=1/2 hence the first step is all one needs...
Jun
28
comment Oscillation - atoms
I see. Do not mortify yourself too much about this, even if you go past 15 minutes... :-)
Jun
28
comment Oscillation - atoms
Late response = 43 minutes.
Jun
28
comment Expected value of discrete functions.
The post does not say that. The OP explains exactly what they mean by "differentiate", a term put between quotes because there is no actual differentiation involved here.
Jun
28
comment How to find the expected cost of an exponential probability?
If your call is 1 minute long, you receive 1 dollar? Cool.
Jun
28
comment What distribution has $X^n$ if $X$ is normal distributed?
How do you compute the distribution of G(X), say when G is monotonous and one knows the distribution of X?
Jun
28
comment How to compute $\mathbb{E}(X \cdot Y \ \textbf{1}_{X \cdot Y \leq n})$?
No, try elementary examples, say X and Y uniform on {1,2} and n=2.
Jun
28
comment $\sup B_t$ has the same distribution as $\sup C_t$ for two brownian motions $B_t, C_t$
Even if B and C are not independent.
Jun
28
comment If $\{x_n\}$ and $\{y_n\}$ are two bounded sequences then prove that $\liminf x_n -\liminf y_n \leq \liminf(x_n - y_n)$
No I will not provide an explicit counterexample since I happen to be convinced that spending five minutes on the hint I gave in my last comment would provide one. Bis: Use @.
Jun
28
comment Quadratic recurrence inequality
Once again, $r_{k+1} \leq r_k^2+ (1/2)r_k$ and $0<r_1<1$ do NOT imply convergence.
Jun
28
comment summation problem
Hint: Write this as $$\sum\limits_{i,j}^{\infty}\alpha^{i}\alpha^{j} -\sum\limits_{i}^{\infty}\alpha^{i}\alpha^{i}$$
Jun
28
comment Let $X$ be a continuous variable with probability density function $kx(1-x)^2$ for $ 0<x<1$
By definition (since $0.1<0.25$), $\mathbb{P}\left[0.25 < X < 0.75\mid X > 0.1\right]=\mathbb{P}\left[0.25 < X < 0.75\right]/\mathbb P\left[X > 0.1\right]$