Search Results

Perhaps its easiest to first ask for the probability $P_0$ that the card does not move at all ($k=0$). One way to achieve this is to cut immediately below and above that card, with probability $1/2\ti …

well, to find a "natural way" to distribute the coefficients $b,c$ in the plane, you could treat this problem as the special case $n=2$ of a classic problem in random-matrix theory: take an $n\times n …

This is a variation of the combinatorial problem considered in section 5 of Some new aspects of the coupon collector’s problem (2003). The $t$ singleton sides (sides which appear once) can be chosen …

I would think the answer is "no" to both questions. Let me abbreviate $v(t)=du/dt$, so $u(t)=u(0)+\int_0^t v(t')dt'$. Now ask for the expectation of $u^2(t)$. You'll need to know how $v(t')$ and $v( …

arXiv:1512.08159 shows that the distribution of the scalar Schur complement in a noncentral Wishart matrix is a mixture of central chi-square distributions with different degrees of freedom. For the c …

Birthday problem: The probability density function of the number of distinct values $Z$ for population size $M$ and sample size $N$ is given by $$P(Z = j) = \binom{M}{j} \sum_{k=0}^j (-1)^k \binom{j} …

The cumulative distribution $F(y)$ of $Y$ is the product of the cumulative distributions $f(x)$ of $X$, because $$F(y)=P[(X_1<y)\cap(X_2<y)\cap\cdots\cap(X_n<y)]=\{f(y)\}^n.$$ Hence $$F(y)=2^{-n}e^{(n …

C. Palm's theory of spatial point processes relies heavily on measure theory in an abstract setting. A more gentle introduction is given in the lecture notes Conditioning in spatial point processes. S …

The complementary binomial distribution is defined by $$\Phi(a;n,p)=\sum _{j=a}^n \binom{n}{j} p^j(1-p)^{n-j},\;\;0<p<1,\;\;0\leq a\leq n.$$ see The derivation of diffusion-jump modes for power plant …

Q: Does anyone know any more information about this theorem? Does it have a name? Why is it true? This result is simply referred to as "Meier's theorem" in the literature. It follows from the property …

All of this relies on the isotropy of the uniform distribution of a vector on the unit sphere. Align your coordinate axis with the unit vector $v$, so that $v\cdot a=a_1$. Then use that $\mathbb{E}[a_ …

The mean of the matrix elements of $Y$ is zero, because the matrix elements of $Q$ and $K$ average to zero. The variance is given by $$\mathbb{E}[Y_{ip}^2]=\sum_{jkl\alpha\beta\gamma}\mathbb{E}[X_{ij} …

$\bullet$ The answer to the first question is affirmative, but the notion of "independence" needs to be modified if we allow for the point process to be directed by an external random variable. A Pois …

Shrirang Mare (2013) gives a proof of Polya's theorem by formulating it as an electric circuit problem and using Rayleigh’s short-cut method from the classical theory of electricity. A similar proof w …

The precise answer depends on the initial condition, let's assume you start the Brownian motion (with diffusion constant $D$) at time $t=0$ in some point $x_0\in(a,b)$, then you ask for the probabilit …