This is part of a short series on the common life data distributions.

The Binomial distribution is discrete. This short article focuses on 4 formulas of the Binomial Distribution.

It has the essential formulas that you may find useful when answering specific questions. Knowing a distribution’s set of parameters does provide, along with the right formulas, a quick means to answer a wide range of reliability related questions.

## Assumptions

Given a count variable and if the following conditions apply then the binomial distribution is rather useful.

- There a fixed number, n, of observations
- The observations are independent
- The outcome of each observation is either success or failure
- The probability of success, p, is the same for each observation

The binomial distribution describes the count variable which is the result of n Bernoulli trials. The occurrence of successes are not ordered thus may occur at any point in the n trials. Thus the use of combinations and not permutations. This assumes replacement or essentially resetting the situation such that the probability, p remains constant.

If we need to assume without replacement consider using the hypergeometric distribution, instead.

## Parameters

The number of trials, n, is fixed and discrete, n ∈ { 0, 1, 2, …, n }

The probability of success, p, also known as the Bernoulli probability parameter is likewise fixed and ranges 0 ≤ p ≤ 1

The count of success, k is a random variable and is count data, k ∈ { 0, 1, 2, …, n }

## Probability Density Function (PDF)

When *t ≥ 0* then the probability density function formula is:

$$ \displaystyle\large f\left( k \right)=\left( \begin{array}{l}n\\k\end{array} \right){{p}^{k}}{{\left( 1-p \right)}^{n-k}}$$

A plot of the PDF provides a histogram-like view of the time-to-failure data.

## Cumulative Density Function (CDF)

*F(t)* is the cumulative probability of failure given k successes. Very handy when estimating the proportion of units that will fail over a warranty period, for example. If each trial represented the warranty period duration of stresses.

$$ \displaystyle\large F\left( k \right)=\sum\limits_{j=0}^{k}{\frac{n!}{j!\left( n-j \right)!}{{p}^{j}}{{\left( 1-p \right)}^{n-j}}}$$

The binomial CDF is a tedious set of calculations and without the benefits of modern computing power has been estimated using Poisson or Normal distribution approximations.

If n ≥ 20 and p ≤ 0.005, or if n ≥ 100 and np ≤ 10, you may use the Poisson distribution with μ = np

$$ \displaystyle\large F\left( k \right)\cong {{e}^{-\mu }}\sum\limits_{j=0}^{k}{\frac{{{\mu }^{j}}}{j!}}$$

If np ≥ 10 and np(1-p) ≥ 10 than the normal distribution provides a suitable approximation

$$ \displaystyle\large F\left( k \right)\cong \Phi \left( \frac{k+0.5-np}{\sqrt{np\left( 1-p \right)}} \right)$$

## Reliability Function

*R(t)* is the chance of k successes. Instead of looking for the proportion that will fail the reliability function determine the proportion that are expected to survive.

$$ \displaystyle\large \begin{array}{l}R\left( k \right)=1-\sum\limits_{j=0}^{k}{\frac{n!}{j!\left( n-j \right)!}{{p}^{j}}{{\left( 1-p \right)}^{n-j}}}\\R\left( k \right)=\sum\limits_{j=k+1}^{n}{\frac{n!}{j!\left( n-j \right)!}{{p}^{j}}{{\left( 1-p \right)}^{n-j}}}\end{array}$$

## Hazard Rate

This is the instantaneous probability of success for a given number of successes, k.

$$ \displaystyle\large \begin{array}{l}h\left( k \right)={{\left[ 1+\frac{{{\left( 1+\theta \right)}^{n}}-\sum\limits_{j=0}^{k}{\left( \begin{array}{l}n\\k\end{array} \right){{\theta }^{j}}}}{\left( \begin{array}{l}n\\k\end{array} \right){{\theta }^{k}}} \right]}^{-1}}\\\text{where}\\\theta =\frac{p}{1-p}\end{array}$$

Mark Liao says

Probably an example will be much helpful for a freshman (like me) to understand besides the formulas.