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As marcoromito wrote, this is an elementary calculation. However, I thought I would record a nice approximation that I stumbled across. Whether it is new, I have no idea. ADDED: The following sente …

Without knowing an answer to the question, I will note the sub-problem of when $W_X(t)$ is a non-increasing function of $t$. Even though it might appear obvious that truncating a distribution makes t …

Here is one approach you can use. The differences $a_k$ provide you with bounds on the difference between the characteristic functions of the two distributions. This may be easy or not depending on …

This paper is relevant: E. Mailhot, Une propriété de la variance de certaines lois de probabilité réelles tronqées, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) 241–244. Mailhot proves that the va …

The case $n=2$ is solved by Charles E. Clark, The Greatest of a Finite Set of Random Variables, Operations Research, Vol. 9, No. 2 (1961), pp. 145-162. In that case the expected maximum is greatest wh …

For a non-negative random variable $X$ whose expectation exists, $$ E(X) = \int_0^\infty \mathrm{Pr}(X>t) dt.$$ In the case of a non-negative integer random variable, this reduces to $$ E(X) = \sum_{i …

$$\prod_{i=1}^{k-1} \,\Bigl(1 - \frac in\Bigr) = \exp\biggl(\sum_{i=1}^{k-1}\ln\Bigl(1-\frac in\Bigr)\biggr).$$ Now use Taylor's theorem to bound $C(n,k)$ such that $$-\frac in -C(n,k)\frac {i^2}{n^ …

My paper here (Adv. Appl. Prob., 21 (1989) 475-478), Theorem 2, provides an estimate over all values of the parameters with relative error that is $o(1)$ if either $\sigma\to\infty$ or $x\sigma\to\inf …

Restricting $Y$ to an annulus doesn't seem useful as any bounds are likely to be satisfied also inside the annulus. A bound with $Y$ restricted to a disk, or more generally to a region with bounded di …

I'm going to change notation a little, using $a_{jk}$ instead of $a_i$ for the vertex $i$ with $r(i)=j$ and $s(i)=k$, and $x_i$ instead of $b_i$. The objective function is $$ J(\boldsymbol{x}) = \sum …

This is called the generalized matrix scaling problem and several other names. Both the theory and associated algorithmic problems have been studied. I suggest you start with this paper and the paper …

Write the probability as $k^{-2n}Q(k,n)$, where $Q(k,n)$ is the sum of the squares of the multinomial coefficients. With the help of OEIS we find that these have been studied before: k=2: http://oe …

There is a large literature on variations of Azuma's inequality. One lemma that is similar to what you ask is Lemma 3.1 of this old paper of Wormald and myself. It considers the case where $|X_k-X_{k- …

Steven's example is indeed the simplest. See chapter 3 of this book for counterexamples to lots of similar possibilities.

This type of discretized normal distribution occurs in some practical problems. For example, the median of the vertex degrees of an Erdős-Renyi random graph with fixed edge probability has such a dis …