Mathematics | Weibull Distribution Model
Last Updated :
19 Feb, 2021
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
- The mean lifetime of these batteries;
- 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.
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