# Probability problem: upper bounds on binomial CDF

• simba31415
In summary, the conversation discusses a problem involving a random variable X and its distribution. The goal is to show that the probability of X being less than a certain constant is less than or equal to a given value. The person asking for help has been advised to use "Chernoff-esque bounds" and has attempted to do so but has not been successful. They suspect that the key to solving the problem lies in showing that the probability can be expressed in terms of the expected value of X. They are seeking assistance in finding a solution to the problem.
simba31415

## Homework Statement

Hi all, just a quick question here - the setup is as follows: X is a random variable, $X \sim \operatorname{Bin}(m,p)$ where $p=2^{-\sqrt{\log n}}(\log n)^2$ and $m \geq 2^{\sqrt{\log n}}c$ for constants c, n (n "large" here). I wish to show that $\mathbb{P}(X < c) \leq e^{-(\log n)^2 c/3}$. I've been told to use "Chernoff-esque bounds" here; however, after teaching myself a little about Chernoff bounds I haven't found a way to make this work - I can see that the multiplicative form could be useful but I haven't yet figured out how to translate the bounds which I've found online into a workable form for this problem.

I'm told observing the fact that $\mu = \mathbb{E}(X) \geq (\log n)^2 c = \mu '$ should also help, so I suspect maybe what we really need is to show is $\mathbb{P}(X < c) = \mathbb{P}(X < \frac{\mu'}{(\log n) ^2}) < \mathbb{P}(X < \frac{\mu}{(\log n) ^2}) \leq ^{(*)} e^{- \mu / 3} \leq e^{-\mu ' /3}$ but as I said, no luck so far since I can't prove step $(*)$ if indeed that is the way to do it. I suspect that this result only needs a few lines of work once you have the bound you require from Chernoff, so if anyone could show me how to do this I'd be very grateful! -S

No thoughts anyone? Sorry to bump but I could really use some help with this, any thoughts at all please respond!

## What is a binomial CDF?

A binomial CDF (cumulative distribution function) is a mathematical function used to determine the probability of obtaining a certain number of successes in a fixed number of independent trials with a known probability of success.

## What is an upper bound?

An upper bound is the maximum possible value that a variable or function can take. In the context of a binomial CDF, it represents the highest possible probability of obtaining a certain number of successes.

## How is an upper bound calculated for a binomial CDF?

The upper bound for a binomial CDF can be calculated by using the binomial distribution formula and plugging in the desired number of trials, probability of success, and number of successes. Alternatively, it can also be found using statistical software or online calculators.

## Why is it important to have an upper bound for a binomial CDF?

Having an upper bound for a binomial CDF can help in decision-making and risk assessment. It allows us to determine the maximum likelihood of obtaining a certain number of successes, which can inform our choices and actions.

## What are some real-world applications of using upper bounds on binomial CDF?

Upper bounds on binomial CDF can be used in various fields such as finance, medicine, and engineering. For example, it can help in calculating the probability of a stock reaching a certain price point, the likelihood of a medical treatment being successful, or the chances of a product failing during stress testing.

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