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4 votes
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Log of a truncated binomial

Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
Tom Solberg's user avatar
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1 vote
0 answers
370 views

Lower bound on the sum of the product of random variables

Let $X_i$ be the $i$-th element of the vector $X=(X_1, ..., X_m)$ of i.i.d. random variables. I am looking for a lower bound for the expression $\mathbb{P}((\sum^n_{i=1}\prod^{m_i}_{j=1}(X_j))^2 \geq ...
Scriddie's user avatar
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1 vote
0 answers
234 views

"Tails" of a multinomial distribution

Let $X_1,\dots,X_N$ denote a collection of independent samples of a uniform multinomial random variable in $\mathbb{Z}^k$, with the number of trials equal to $n\ll k$. (By "uniform", I mean ...
Tom Solberg's user avatar
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106 views

Upper bounding the sum with hypergeometric and binomial probabilities

Could you please help me upper bound this tricky expression: $$P(A)=\sum_{i=0}^n{\left( 1 - \dfrac{\binom kq \binom {n-k}{i-q}}{\binom {n}{i}} \right)}^I \binom ni p^i {(1-p)}^{n-i}$$. So far I only ...
abs135's user avatar
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