### Maths Weibull Distribution

##### Weibull Distribution

The Weibull distribution is a continuous probability distribution that describes the distribution of time until a specified event occurs. It is named after the Swedish mathematician Waloddi Weibull, who first proposed it in 1939.

##### Characteristics of the Weibull Distribution

The Weibull distribution is characterized by two parameters: the scale parameter $\lambda$ and the shape parameter $k$. The scale parameter determines the spread of the distribution, while the shape parameter determines the skewness of the distribution.

**Scale parameter $\lambda$**: The scale parameter $\lambda$ is a positive real number that determines the spread of the distribution. The larger the value of $\lambda$, the more spread out the distribution will be.**Shape parameter $k$**: The shape parameter $k$ is a positive real number that determines the skewness of the distribution. When $k < 1$, the distribution is skewed to the left. When $k = 1$, the distribution is symmetric. When $k > 1$, the distribution is skewed to the right.

##### Probability Density Function of the Weibull Distribution

The probability density function (PDF) of the Weibull distribution is given by:

$$f(x) = \frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k}$$

where:

- $x$ is the random variable
- $\lambda$ is the scale parameter
- $k$ is the shape parameter

##### Cumulative Distribution Function of the Weibull Distribution

The cumulative distribution function (CDF) of the Weibull distribution is given by:

$$F(x) = 1 - e^{-(x/\lambda)^k}$$

where:

- $x$ is the random variable
- $\lambda$ is the scale parameter
- $k$ is the shape parameter

##### Applications of the Weibull Distribution

The Weibull distribution is used in a variety of applications, including:

- Reliability engineering: The Weibull distribution is used to model the distribution of time until failure for components and systems.
- Survival analysis: The Weibull distribution is used to model the distribution of time until death for individuals in a population.
- Financial modeling: The Weibull distribution is used to model the distribution of returns on investments.
- Insurance: The Weibull distribution is used to model the distribution of claims.

The Weibull distribution is a versatile and powerful probability distribution that can be used to model a wide variety of phenomena. It is a valuable tool for statisticians and practitioners in a variety of fields.

##### Formula of Weibull Distribution

The Weibull distribution is a continuous probability distribution that describes the distribution of time until a specified event occurs. It is named after the Swedish mathematician Waloddi Weibull, who first proposed it in 1939.

The Weibull distribution is often used to model the lifetime of components or systems, as it can account for both the initial “infant mortality” period and the subsequent “wear-out” period.

##### Probability Density Function

The probability density function (PDF) of the Weibull distribution is given by:

$$f(x) = \frac{\beta}{\alpha} \left(\frac{x}{\alpha}\right)^{\beta-1} e^{-(x/\alpha)^{\beta}}$$

where:

- $\alpha$ is the scale parameter, which represents the characteristic life of the distribution.
- $\beta$ is the shape parameter, which determines the shape of the distribution.

##### Cumulative Distribution Function

The cumulative distribution function (CDF) of the Weibull distribution is given by:

$$F(x) = 1 - e^{-(x/\alpha)^{\beta}}$$

##### Hazard Function

The hazard function of the Weibull distribution is given by:

$$h(x) = \frac{\beta}{\alpha} \left(\frac{x}{\alpha}\right)^{\beta-1}$$

##### Example

Suppose we have a component that has a Weibull distribution with $\alpha = 100$ and $\beta = 2$. The probability that the component will fail within the first 50 hours is given by:

$$P(X < 50) = 1 - e^{-(50/100)^2} = 0.197$$

This means that there is a 19.7% chance that the component will fail within the first 50 hours.

##### Properties of Weibull Distribution

The Weibull distribution is a continuous probability distribution that is often used to model the distribution of failure times in reliability engineering. It is named after the Swedish mathematician Waloddi Weibull, who first proposed it in 1939.

##### Probability Density Function

The probability density function (PDF) of the Weibull distribution is given by:

$$f(x) = \frac{\beta}{\alpha} \left(\frac{x}{\alpha}\right)^{\beta-1} e^{-(x/\alpha)^{\beta}}$$

where:

- $\alpha$ is the scale parameter
- $\beta$ is the shape parameter

##### Cumulative Distribution Function

The cumulative distribution function (CDF) of the Weibull distribution is given by:

$$F(x) = 1 - e^{-(x/\alpha)^{\beta}}$$

##### Shape of the Weibull Distribution

The shape of the Weibull distribution can vary depending on the values of the scale and shape parameters.

- When $\beta < 1$, the distribution is decreasing failure rate (DFR).
- When $\beta = 1$, the distribution is an exponential distribution.
- When $\beta > 1$, the distribution is increasing failure rate (IFR).

##### Mean of the Weibull Distribution

The mean of the Weibull distribution is given by:

$$E(X) = \alpha \Gamma\left(1 + \frac{1}{\beta}\right)$$

where $\Gamma(\cdot)$ is the gamma function.

##### Variance of the Weibull Distribution

The variance of the Weibull distribution is given by:

$$V(X) = \alpha^2 \left[\Gamma\left(1 + \frac{2}{\beta}\right) - \Gamma^2\left(1 + \frac{1}{\beta}\right)\right]$$

##### Weibull Parameters

The Weibull distribution is a continuous probability distribution that is used to model the time to failure of a component or system. It is named after the Swedish mathematician Waloddi Weibull, who first proposed it in 1939.

The Weibull distribution has two parameters:

**Scale parameter (β):**The scale parameter represents the characteristic life of the component or system. It is the value of the random variable at which the cumulative distribution function (CDF) is equal to 0.632.**Shape parameter (α):**The shape parameter represents the shape of the distribution. It determines the rate at which the failure rate increases over time.

##### Scale Parameter (β)

The scale parameter (β) is a measure of the central tendency of the Weibull distribution. It is the value of the random variable at which the CDF is equal to 0.632. This means that the probability of failure before the scale parameter is 0.632, and the probability of failure after the scale parameter is 0.368.

The scale parameter is often estimated using the median of the failure data. The median is the value of the random variable that divides the data into two equal parts. For the Weibull distribution, the median is given by:

$$Median = β * (ln 2)^{1/α}$$

##### Shape Parameter (α)

The shape parameter (α) is a measure of the dispersion of the Weibull distribution. It determines the rate at which the failure rate increases over time. A small value of α indicates that the failure rate increases slowly over time, while a large value of α indicates that the failure rate increases rapidly over time.

The shape parameter is often estimated using the slope of the log-log plot of the failure data. The log-log plot is a plot of the logarithm of the failure time versus the logarithm of the cumulative failure probability. For the Weibull distribution, the log-log plot is a straight line with a slope of α.

##### Weibull Distribution Plot

The Weibull distribution is a continuous probability distribution that is used to model the distribution of failure times. It is a generalization of the exponential distribution and is named after the Swedish mathematician Waloddi Weibull.

##### Characteristics of the Weibull Distribution

The Weibull distribution has a number of characteristics that make it useful for modeling failure times. These characteristics include:

- The Weibull distribution is a
**unimodal**distribution, meaning that it has a single peak. - The Weibull distribution is a
**skewed**distribution, meaning that the tail of the distribution is longer on one side than the other. - The Weibull distribution has a
**scale parameter**and a**shape parameter**. The scale parameter determines the spread of the distribution, while the shape parameter determines the skewness of the distribution.

##### Plotting the Weibull Distribution

The Weibull distribution can be plotted using a variety of methods. One common method is to use a **Weibull probability plot**. A Weibull probability plot is a graphical representation of the cumulative distribution function (CDF) of the Weibull distribution.

To create a Weibull probability plot, the following steps are followed:

- The data is sorted in ascending order.
- The CDF of the data is calculated.
- The CDF is plotted against the natural logarithm of the data.

The resulting plot will be a straight line if the data follows the Weibull distribution.

The Weibull distribution is a useful tool for modeling the distribution of failure times. It is a flexible distribution that can be used to model a variety of different types of data.

##### Solved Examples

##### Example 1: Finding the Area of a Circle

**Problem:** Find the area of a circle with radius 5 cm.

**Solution:**

- The formula for the area of a circle is $A = \pi r^2$, where $A$ is the area, $r$ is the radius, and $\pi$ is a mathematical constant approximately equal to 3.14.
- Substituting $r = 5$ cm into the formula, we get $A = \pi (5)^2 = 25\pi$ cm$^2$.
- Therefore, the area of the circle is $25\pi$ cm$^2$.

##### Example 2: Solving a Linear Equation

**Problem:** Solve the linear equation $3x + 5 = 17$.

**Solution:**

- Subtract 5 from both sides of the equation: $3x + 5 - 5 = 17 - 5$.
- Simplify: $3x = 12$.
- Divide both sides of the equation by 3: $\frac{3x}{3} = \frac{12}{3}$.
- Simplify: $x = 4$.
- Therefore, the solution to the linear equation is $x = 4$.

##### Example 3: Finding the Volume of a Cube

**Problem:** Find the volume of a cube with side length 4 cm.

**Solution:**

- The formula for the volume of a cube is $V = s^3$, where $V$ is the volume and $s$ is the side length.
- Substituting $s = 4$ cm into the formula, we get $V = (4)^3 = 64$ cm$^3$.
- Therefore, the volume of the cube is $64$ cm^3.

##### Example 4: Solving a Quadratic Equation

**Problem:** Solve the quadratic equation $x^2 - 5x + 6 = 0$.

**Solution:**

- We can use the quadratic formula to solve this equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.
- In this case, $a = 1$, $b = -5$, and $c = 6$. Substituting these values into the formula, we get:

$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)}$$

$$= \frac{5 \pm \sqrt{25 - 24}}{2}$$

$$= \frac{5 \pm 1}{2}$$

- Therefore, the solutions to the quadratic equation are $x = 2$ and $x = 3$.

##### Example 5: Finding the Derivative of a Function

**Problem:** Find the derivative of the function $f(x) = x^3 - 2x^2 + 3x - 4$.

**Solution:**

- The derivative of a function is the rate of change of the function with respect to its input.
- To find the derivative of $f(x)$, we can use the power rule of differentiation, which states that the derivative of $x^n$ is $nx^{n-1}$.
- Applying the power rule, we get:

$$f’(x) = \frac{d}{dx}(x^3 - 2x^2 + 3x - 4)$$

$$= 3x^2 - 4x + 3$$

- Therefore, the derivative of $f(x)$ is $3x^2 - 4x + 3$.

##### Weibull Distribution FAQs

##### What is the Weibull distribution?

The Weibull distribution is a continuous probability distribution that describes the distribution of time until failure in a system. It is named after the Swedish mathematician Waloddi Weibull, who first proposed it in 1939.

##### What are the parameters of the Weibull distribution?

The Weibull distribution has two parameters:

**Scale parameter**(λ): This parameter represents the characteristic life of the system. It is the time at which the probability of failure is 63.2%.**Shape parameter**(k): This parameter represents the shape of the distribution. A value of k < 1 indicates a decreasing failure rate, a value of k = 1 indicates a constant failure rate, and a value of k > 1 indicates an increasing failure rate.

##### What are the properties of the Weibull distribution?

The Weibull distribution has a number of properties that make it useful for modeling time-to-failure data. These properties include:

- The Weibull distribution is a unimodal distribution, meaning that it has a single mode.
- The Weibull distribution is a skewed distribution, meaning that the tail of the distribution is longer on one side than the other.
- The Weibull distribution is a heavy-tailed distribution, meaning that the probability of a large failure time is greater than the probability of a small failure time.
- The Weibull distribution is a versatile distribution that can be used to model a wide variety of failure time data.

##### How is the Weibull distribution used?

The Weibull distribution is used in a variety of applications, including:

- Reliability engineering: The Weibull distribution is used to model the time to failure of components and systems.
- Quality control: The Weibull distribution is used to monitor the quality of products and processes.
- Survival analysis: The Weibull distribution is used to model the survival times of patients and other individuals.
- Insurance: The Weibull distribution is used to model the time to claim for insurance policies.

##### What are some of the limitations of the Weibull distribution?

The Weibull distribution has a number of limitations, including:

- The Weibull distribution is not always the best distribution to model time-to-failure data. There are other distributions that may be more appropriate for certain types of data.
- The Weibull distribution can be difficult to fit to data, especially when the data is censored.
- The Weibull distribution can be sensitive to outliers.

##### Conclusion

The Weibull distribution is a powerful tool for modeling time-to-failure data. However, it is important to understand the limitations of the distribution before using it.