All Questions
9 questions with no upvoted or accepted answers
5
votes
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205
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Strange inequality relating Binomial pmf and cdf
I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
- 101
5
votes
0
answers
523
views
How to obtain the probability distribution of a sum of dependent discrete random variables more efficiently
I hope you are well. Here is my problem.
Let $\{s_0,\,s_1,\ldots,\,s_T\}$ be a sequence of discrete random variables and denote $S_t=s_0+s_1+\cdots+s_t$, with $S_0=0$ and $S_T\leq M$, where $M$ and $T$...
- 173
4
votes
0
answers
131
views
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 $$\frac{\boldsymbol{E}\log\...
- 4,049
4
votes
0
answers
823
views
Total Variation distance of polynomials of Bernoulli R.V.s
Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with
$\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$.
Let
\begin{align*}
X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...
- 175
4
votes
0
answers
516
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Sum of Binomial random variable CDF
Suppose there are two independent Binomial random variables
$$
X\sim Binomial(n,p)\\
Y\sim Binomial(n,p+\delta)
$$
where $\delta$ is considered to be fixed, and $p$ can vary in $(0,1-\delta)$.
Now ...
- 103
2
votes
0
answers
77
views
Inequalities concerning cummulative distributions of binomials
For random variable $Z$, let $F_Z$ denote its cdf, i.e., $F_Z(t)=\mathbb{P}(Z\leq t)$. Let $X$ be a binomial distribution with parameters $(n,p)$ and $Y$ a binomial distribution with parameters $(m,p)$...
- 121
1
vote
0
answers
126
views
Quotient of cumulative binomial distribution functions
Given to integers $n < m \in \mathbb{N}_0$ and a probability $p$, I'm struggling to calculate (or at least get an upper bound for) the quotient
$$Q = \frac{F(n+1;m,p)}{F(n;m,p)}$$
where $F$ denotes ...
- 11
0
votes
0
answers
159
views
How to express the expectation and variance of a truncated binomial distribution without summation?
Given a binomial distribution with parameters $ n $ and $ p $, where $ n $ is an odd integer greater than or equal to 3, I am interested in the truncated binomial distribution where we truncate at $ k ...
- 1
0
votes
0
answers
102
views
Lower bound for the probability that $X=\omega\left(\mathbb E[X]\right)$ for $X\sim Bin(n,p)$
Let $X\sim Bin(n,p)$ be a binomial variable and let $\delta\in (0,1)$.
I'm looking for a lower bound of the form $\Pr[X > f(\delta)] \ge \delta$.
Specifically, if $\delta,p=o(1)$ are not ...
- 618