The 4 Parameter Beta Distribution 7 Formulas

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

The Beta distribution is a univariate continuous distribution. This short article focuses on 7 formulas of the Beta Distribution.

If you want to know more about fitting a set of data to a distribution, well that is in another article.

The Beta function is not used to describe life data very often yet is used to describe model parameters that are contained within an interval. For example given a probability parameter constrained from 0 ≤ p ≤ 1 the use of the Beta distribution is well suited to model such a parameter.

The Beta distribution is also known as a Pearson Type I distribution.

Parameters

The shape parameter, α, is always greater than zero. As is the second shape parameter, β, also always great then zero

The location parameter, known as the lower bound, a_L ranges from -∞ < a_L < b. For a standard Beta distribution, a_L = 0.

The final parameter, known as the upper bound, b_U ranges from a < b_U < ∞. For a standard Beta distribution, b_U = 1.

We may also define the scale parameter as b – a.

The Beta distribution has limits of a_L < t ≤ b_U.

• Let’s define:
• B(x,y) as the Beta function.
• B_t(t|x,y) as the incomplete Beta function.
• I_t(t|x,y) as the regularized Beta function.
• Γ(k) is the complete gamma

One interesting characteristic is B(α,β) is the mirror distribution of B(β,α) Thus, if X random variable is B(α,β), and if Y = 1 – X then Y is B(β,α).

Probability Density Function (PDF)

When a_L < t ≤ b_U then the probability density function formula is:

$$ \displaystyle\large f\left( t;\alpha ,\beta ,{{a}_{L}},{{b}_{U}} \right)=\frac{\Gamma \left( \alpha +\beta \right)}{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}\bullet \frac{{{\left( t-{{a}_{L}} \right)}^{\alpha -1}}{{\left( {{b}_{U}}-1 \right)}^{\beta -1}}}{{{\left( {{b}_{U}}-{{a}_{L}} \right)}^{\alpha +\beta -1}}}$$

Or for the standard Beta distribution,

$$ \displaystyle\large \begin{array}{l}f\left( t|\alpha ,\beta \right)=\frac{\Gamma \left( \alpha +\beta \right)}{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}\bullet {{t}^{\alpha -1}}{{\left( 1-t \right)}^{\beta -1}}\\f\left( t|\alpha ,\beta \right)=\frac{1}{B\left( \alpha ,\beta \right)}\bullet {{t}^{\alpha -1}}{{\left( 1-t \right)}^{\beta -1}}\end{array}$$

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 from time zero till time t. Very handy when estimating the proportion of units that will fail over a warranty period, for example.

$$ \displaystyle\large \begin{array}{l}F\left( t \right)=\frac{\Gamma \left( \alpha +\beta \right)}{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}\int_{0}^{t}{{{u}^{\alpha -1}}\left( 1-t \right){{u}^{\beta -1}}du}\\F\left( t \right)=\frac{B\left( t|\alpha ,\beta \right)}{B\left( \alpha ,\beta \right)}\\F\left( t \right)=I\left( t|\alpha ,\beta \right)\end{array}$$

Reliability Function

R(t) is the chance of survival from from time zero till time t. Instead of looking for the proportion that will fail the reliability function determine the proportion that are expected to survive.

$$ \displaystyle\large R\left( t \right)=1-I\left( t|\alpha ,\beta \right)$$

Conditional Survivor Function

The m(x) function provides a means to estimate the chance of survival for a duration beyond some known time, t, over which the item(s) have already survived. What the probability of surviving time x given the item has already survived over time t?

$$ \displaystyle\large m\left( t \right)=R(x|t)=\frac{1-I\left( t+x|\alpha ,\beta \right)}{1-I\left( t|\alpha ,\beta \right)}$$

Mean Residual Life

This is the cumulative expected life over time x given survival till time t.

$$ \displaystyle\large u\left( t \right)=\frac{\int_{t}^{\infty }{\left\{ B\left( \alpha ,\beta \right)-{{B}_{x}}\left( x|\alpha ,\beta \right) \right\}dx}}{B\left( \alpha ,\beta \right)-{{B}_{t}}\left( t|\alpha ,\beta \right)}$$

Hazard Rate

This is the instantaneous probability of failure per unit time.

$$ \displaystyle\large h\left( t \right)=\frac{{{t}^{\alpha -1}}\left( 1-t \right)}{B\left( \alpha ,\beta \right)-{{B}_{t}}\left( t|\alpha ,\beta \right)}$$

Cumulative Hazard Rate

This is the cumulative failure rate from time zero till time t, or the area under the curve described by the hazard rate, h(x).