# 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

- 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.