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A Gaussian Lower Bound for a Binomial Right Tail

We show that for all integers $k \ge 1$ and all $p \in (0, \tfrac{1}{2})$, if

$$n := 2k,\qquad \sigma^2 := p(1-p),\qquad z := \frac{(1/2 - p)\sqrt{2k}}{\sigma},$$

then

$$1 - \Phi(z) + \frac{1}{2} \binom{2k}{k},\sigma^{2k} ;\le; \mathbb{P}\bigl(\mathrm{Bin}(2k,p)\ge k\bigr),$$

winger

import Mathlib.Data.Nat.Basic
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Order.Filter.AtTopBot.Basic

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import Mathlib.Data.List.Sort
import Mathlib.Data.List.Basic
import Mathlib.Data.Nat.Digits.Defs
import Mathlib.Algebra.BigOperators.Group.Finset.Defs
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Finset.Sort
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.List

winger

import torch
import numpy as np
from matplotlib import pyplot as plt

torch.set_printoptions(precision=10)

class ResidualBlock(torch.nn.Module):
    def __init__(self, dims, bottleneck):
        super(ResidualBlock, self).__init__()
        self.linear1 = torch.nn.Linear(dims, bottleneck)

winger

#pragma once
#include <iostream>
#include <algorithm>
#include <vector>
#include <unordered_map>
#include <string>
#include <variant>

using Key = std::string;
using Value = std::variant<std::string, double, bool>;

winger

root = "cargo"

[packages]

[packages.bitflags]
dependencies = []
path = "/Users/winger/.cargo/registry/src/github.com-1ecc6299db9ec823/bitflags-0.1.1"
version = "0.1.1"

[packages.cargo]

winger

s = input()

prevs = dict()
a = []
for i in range(len(s)):
    if s[i] in prevs:
        a.append(i - prevs[s[i]])
    else:
        a.append(i + 1)
    prevs[s[i]] = i