Binomial distribution¶

Describes the number of successes out of total n Bernoulli trials with a set probability. Where it is bounded \((0,n)\) where n is the total number of trials.

PDF:¶

\[ f(x| n, p) = \binom{n}{x}p^{x}(1-p)^{n-x}\]

n is the total number of trials
p is the probability of success on any given trial
\(\binom{n}{p}\) is the normalizing constant, defines the total number of ways x can be drawn out of n trials

Integral of an binomial distribution¶

\[\int_{0}^1 \binom{n}{k} \theta^k (1-\theta)^{n-k}d\theta = \binom{n}{k} \frac{\Gamma(k+1)\Gamma{(n - k +1)}}{\Gamma{(n+k)}} = \frac{1}{n+1}\]

An alternatively this can be explained using bayes biliard

Properties¶

variance is greatest when \(p=0.5\) and decreases as p gets closer to either 0 or 1.
when n is large and p is away from 0 or 1, it approaches the normal distribution
wne n is large and p is small, it approaches the Poisson distribution

Convergence to Normal distribution¶

Since the Binomial distribution can be viewed as a sum of i.i.d Bern(p) random variable therefore for a large n, we can use the law or large number for approximation:

\[ Y \sim Bin(n,p) \approx N(np, np(1-p))\]

Normal distribution is continuous but the binomial is discrete thus the probability \(P(Y=k)\) would be 0. For this case we need to use the continuity correction:

\[ p(Y=k) = P(k-1/2 < Y < k + 1/2) \approx \Phi(\frac{k _ 1/2 - np}{\sqrt{np(1-p)}}) - \Phi(\frac{k - 1/2 - np}{\sqrt{np(1-p)}}) \]

Connection to Hypergeometric¶

We have 2 Binomial distributions with the same probability \(p\)

\[\begin{split}X \sim \text{Bin}(n,p) \\ Y \sim \text{Bin}(m, p)\end{split}\]

If we assume that \(X \perp Y\) and take the sum of those two random variables \(X + Y = r\). Now we can condition one of the distributions on the sum:

\[P(X|X+Y) = \text{HGeom}(n,m,r) \]

An example: We have \(n\) women and \(m\) men. There is a virus and the probability that a random person is infected is \(p\). Thus the number of infected woman is \(X\sim Bin(n,p)\) and the number of infected men is \(Y\sim Bin(m,p)\). Now we pick a random sample from the population and we want to know what is the probability that a sample size of x of woman is infected.

study-notes