Question

. Let \(Y_{1}

Short answer

The distribution function (CDF) of the smallest order statistic from a Weibull distribution, \(Y_{1}\), is given by \(F_{Y_1}(y_1)=1-e^{-n(y_1/\lambda)^{k}}\) and the probability density function (pdf) is given by \(f_{Y_1}(y_1) = n \frac{k}{\lambda } \left(\frac{y_1}{\lambda }\right) ^{k-1} e^{-n(y_1/\lambda)^{k}}\)

Step-by-step solution

Understand the properties of the Weibull distribution

The Weibull distribution has a pdf represented by \(f(y;\lambda,k) = \frac{k}{\lambda} \left(\frac{y}{\lambda}\right)^{k-1} e^{-(y/\lambda)^{k}} \text{ for } y\geq0, \lambda>0, k>0\). The cumulative distribution function (CDF) is represented by \(F(y;\lambda,k)=1-e^{-(y/\lambda)^{k}} \text{ for } y\geq0\).

Finding the joint distribution of order statistics

The joint distribution of ordered statistics of a random sample of size \(n\) from a population with pdf \(f(y)\) and cumulative distribution function (CDF) \(F(y)\) is \(f(y_1, y_2, ..., y_n)=n!f(y_1)f(y_2)...f(y_n)[F(y_1)...(F(y_2)-F(y_1))...(F(y_n)-F(y_{n-1}))...(1-F(y_n))]\). Here \(y_1<y_2<...<y_n\)

Derive the cumulative distribution function (CDF) of \(Y_{1}\)

The CDF of the minimum order statistic, \(Y_{1}\), is represented by \(F_{Y_1}(y_1)=1-(1-F(y_1))^n\), since \(1-F(y_1)\) represents the probability of a random observation exceeding \(y_1\) and \(1-(1-F(y_1))^n\) is the cumulative probability that at least one observation out of \(n\) does not exceed \(y_1\) (or at least one observation equals the minimum, \(y_1\)). For our specific case of the Weibull distribution, the CDF of \(Y_{1}\) becomes \(F_{Y_1}(y_1)=1-e^{-n(y_1/\lambda)^{k}}\)

Derive the probability density function (pdf) of \(Y_{1}\)

Once we have the CDF, we can find the pdf by taking the derivative of the CDF. In this case, the pdf of \(Y_{1}\) becomes \(f_{Y_1}(y_1) = \frac{d}{dy_1} F_{Y_1}(y_1) = n \frac{k}{\lambda } \left(\frac{y_1}{\lambda }\right) ^{k-1} e^{-n(y_1/\lambda)^{k}}\)

Key concepts

Order Statistics

Order statistics are an important concept in probability and statistics. They involve arranging a set of random variables in increasing order. For instance, if you have a random sample of size \(n\) from a given distribution, the variables \(Y_{1}, Y_{2}, \dots, Y_{n}\) denote these variables ordered such that \(Y_{1} < Y_{2} < \cdots < Y_{n}\). Here, the smallest value is \(Y_{1}\) and the largest is \(Y_{n}\).

• Order statistics help in finding extreme values, such as minimum or maximum.
• They are widely used in reliability analysis and operations research.
• Often utilized in financial risk modeling to identify worst-case scenarios.

In practice, you might be interested in the distribution of the smallest value (like \(Y_{1}\)), since it often holds importance in risk assessments and quality control sectors.

Cumulative Distribution Function

The cumulative distribution function (CDF) is a crucial tool for understanding probabilities. It describes the probability that a random variable takes a value less than or equal to a certain number. For a random variable \(Y\), the CDF is denoted as \(F(y)\) and is defined as:\[F(y) = P(Y \leq y)\]

• The CDF starts from 0 and increases to 1 as you move along the distribution.
• It provides a comprehensive description of all probability aspects of the distribution.
• Useful in computing probabilities within a range and finding quantiles.

In the case of the Weibull distribution, the CDF helps you determine how likely it is for a variable to be less than or equal to a value \(y\). For instance, the CDF of the minimum order statistic, \(Y_{1}\), can be calculated as \(F_{Y_1}(y_1)=1-e^{-n(y_1/\lambda)^{k}}\). This formula reflects the cumulative probability of at least one observation being smaller or equal to a given \(y_1\) in a sample of size \(n\).

Probability Density Function

The probability density function (PDF) is another vital concept that complements the CDF. It provides the likelihood of a continuous random variable to take on a specific value. For a distribution's PDF, symbolized typically as \(f(y)\), it involves:\[f(y) = \lim_{\Delta y \to 0} \frac{P(y \leq Y < y + \Delta y)}{\Delta y}\]

• The PDF is not a probability; it needs to integrate to 1 over the entire distribution.
• For continuous distributions, the PDF at any particular point can be found by differentiating the CDF.
• Useful in recognizing where values are most densely packed or spread out in a distribution.

In our special case, when it comes to the Weibull's smallest order statistic \(Y_{1}\), the process involves getting the probability density by differentiating its CDF. The resulting PDF is expressed as \(f_{Y_1}(y_1) = n \frac{k}{\lambda } \left(\frac{y_1}{\lambda }\right)^{k-1} e^{-n(y_1/\lambda)^{k}}\). This formula reveals how the probability density behaves, emphasizing occurrences of the smallest value in the sample.