The 2 Parameter Binomial Discrete Distribution 4 Formulas

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.

1. There a fixed number, n, of observations
2. The observations are independent
3. The outcome of each observation is either success or failure
4. 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.

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