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If you search at Google Scholar for "sum of lognormal" or "sum of log-normal" (using the quotation marks), you will find several papers devoted to this question.

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^ …

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 …

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 …

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 …

Take a bernoulli variable with weight $\log^{-4} n$ at $-\log n$ and weight $1-\log^{-4} n$ at $\log n/(\log^4 n-1)$. Then the mean is 0, the variance is $O(\log^{-2} n)$, the 4th moment converges to …

I just thought I'd record a cute "answer". $$ \operatorname{Prob}(X-Y=k) = \frac{1}{2\pi} \int_{-\pi}^\pi \cos(k\theta)\,(2P\cos\theta+1-2P)^n\,d\theta, $$ where $P=p(1-p)$. If $k$ is not …

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 …

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

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 a confident guess, without a proof. The normal approximation of the binomial distribution gives the estimate $$ f(n,p) = \frac{\sqrt{2pq}}{\sqrt{\pi n}}.$$ Now, experimentally, for any fixed …

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 …