Mathematics | Weibull Distribution Model

Introduction :

Suppose an event can occur several times within a given unit of time. When the total number of occurrences of the event is unknown, we can think of it as a random variable X. When this random variable X follows Weibull Distribution Model (closely related to the exponential distribution) then its probability density function is given as follows.

only when 
x > 0, α >0, β > 0.  
f(x) = 0 , Otherwise

The cumulative distribution function of Weibull Distribution is obtained as follows.

Putting y = , 

We will get the following expression.

This means, when X has the Weibull distribution then Y =  has an exponential distribution.

Expected Value :

The Expected Value of the Beta distribution can be found by summing up products of Values with their respective probabilities.

Putting u = , 

We will get the following expression as follows.

Using the definition of the gamma function, we will get the following expression as follows.

Variance and Standard Deviation :

The Variance of the Beta distribution can be found using the Variance Formula.

Putting u = , 

We will get the following expression as follows.

Using the definition of the gamma function, we will get the following.

Standard Deviation is given as follows.

Example –

Suppose that the lifetime of a certain kind of emergency backup battery (in hours) is a random variable X having the Weibull distribution with α = 0.1 and β = 0.5. Find

1. The mean lifetime of these batteries;
2. The probability that such a battery will last more than 300 hours.

Solution –

1. Substituting the value of α and β in the formula of mean we will get the following.

hours

2. The probability of the battery lasting for more than 300 hours is given by the following.