What is Cumulative Binomial probabilities?

Question

I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is.

So my question is, What is Cumulative Binomial probabilities ? any example will be of great help.

Thanks! :)

Answer 1

I guess you mean the cumulative distribution function $F(k;n,p)$ of the (discrete) binomial distribution with number of trials $n \ge 0$ and success probability $0 \le p \le 1.\;$ The probability mass functions (PMF) is $$f(k;n,p) = \binom{n}{k}p^k(1-p)^{n-k} $$ and the CDF can be expressed with normalized incomplete Beta function

$$F(k;n,p) = \sum_{i=0}^{k}f(k;n,p) = I_{1-p}(n-k,k+1)$$

Answer 2

I sense that you are implicitly asking what a cumulative density function is. I apologize if this is not the case, and you can ignore what follows.

Imagine that your Binomial random variable represents the number of heads you obtain while flipping some coin $n$ times (let's formally call this random variable $X$). Then, the cumulative density function (or CDF) is a function that tells you, for each natural number $k$, what is the probability that you will obtain at maximum $k$ heads. If your coin is biased and it has a probability of showing heads equal $p$, the definition the CDF is

$F(k) = \mathbb P (X \leq k)$.

This definition is general, it works for all random variables and not only Binomials! For the specific Binomial case, gammatester gave you the correct formula.