Problem 8 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Let \(X\) have an \(\operatorname{Exp}(1)\) distribution, and let \(\alpha\) and \(\lambda\) be positive numbers. Determine the distribution function of the random variable $$ W=\frac{X^{1 / \alpha}}{\lambda} $$ The distribution of the random variable \(W\) is called the Weibull distribution with parameters \(\alpha\) and \(\lambda\).

Step-by-Step Solution

Answer

The distribution function of \(W\) is \(F_W(w) = 1 - e^{-(\lambda w)^\alpha}\), which is the CDF of the Weibull distribution.

Step 1: Define the Distribution of X

The random variable \(X\) with an \(\operatorname{Exp}(1)\) distribution has the probability density function (pdf) given by \(f_X(x) = e^{-x}\) for \(x \geq 0\). The cumulative distribution function (CDF) is \(F_X(x) = 1 - e^{-x}\).

Step 2: Define the Transformation

We are given the transformation \(W = \frac{X^{1/\alpha}}{\lambda}\). We need to find the CDF of \(W\), denoted as \(F_W(w)\).

Step 3: Express CDF of W in Terms of X

The CDF of \(W\) is given by \(F_W(w) = P(W \leq w) = P\left( \frac{X^{1/\alpha}}{\lambda} \leq w \right)\). Simplifying, we have \(X^{1/\alpha} \leq \lambda w\).

Step 4: Solve the Inequality

The inequality \(X^{1/\alpha} \leq \lambda w\) implies \(X \leq (\lambda w)^\alpha\). Thus, \(F_W(w) = P(X \leq (\lambda w)^\alpha)\).

Step 5: Apply the Distribution of X

Since \(X\) has an Exponential(1) distribution, use its CDF to find \(F_W(w) = 1 - e^{-(\lambda w)^\alpha}\). This gives us the CDF of the Weibull distribution with parameters \(\alpha\) and \(\lambda\).

Step 6: Conclusion

The distribution function of the random variable \(W\) is given by \(F_W(w) = 1 - e^{-(\lambda w)^\alpha}\). This matches the CDF of the Weibull distribution with parameters \(\alpha\) and \(\lambda\).

Key Concepts

Exponential Distribution

The exponential distribution is one of the simplest and most widely used probability distributions. It's frequently used to model time until an event happens, like how long you'll wait for your coffee. The time until failure of mechanical systems, like light bulbs, is another classic example.

Here are some key points about the Exponential Distribution:

• The probability density function (pdf) is denoted by \( f_X(x) = e^{-x} \) for \( x \geq 0 \). This shows the rapidly decreasing likelihood as x increases, which is characteristic of exponential decay.
• The cumulative distribution function (CDF), which provides the probability of the random variable being less than a specific value, is given by \( F_X(x) = 1 - e^{-x} \). This means as time (x) increases, the probability of the event occurring approaches 1, simplifying the analysis of time to failure.
• The exponential distribution is determined by a single parameter, often presented as the rate \( \lambda \), when \( X \sim \operatorname{Exp}(\lambda) \). In our problem, \( \lambda = 1 \), simplifying calculations.

Understanding these concepts is crucial when working with models that use exponential distributions, especially in reliability and survival analysis.

Weibull Distribution

The Weibull distribution generalizes the exponential distribution by introducing a shape parameter \( \alpha \), offering greater versatility. Because of its flexibility, it's employed in a wide variety of fields. Let's delve into how it works and why it's so useful.

Here's what makes the Weibull Distribution special:

• It can model various types of data depending on the shape parameter \( \alpha \). For example, when \( \alpha = 1 \), the Weibull distribution resembles the exponential distribution.
• The inclusion of the parameter \( \lambda \), called the scale parameter, adjusts the spread of the distribution. By transforming \( X \) as \( W = \frac{X^{1/\alpha}}{\lambda} \), we effectively scale the data and modify its shape.
• The CDF of the Weibull distribution with parameters \( \alpha \) and \( \lambda \) is given as \( F_W(w) = 1 - e^{-(\lambda w)^\alpha} \), which controls both the scaling and the shape of data.

This distribution provides a robust way to model data that standard exponential distributions might not fit well, crucial for reliability analysis and life data analysis.

Cumulative Distribution Function

The cumulative distribution function (CDF) is a fundamental concept in probability and statistics. It provides a comprehensive way to understand the probability distribution of a random variable.

Here’s why the CDF is important:

• The CDF of a random variable \( X \), such as \( F_X(x) \), shows the probability that \( X \) will take a value less than or equal to \( x \). It aggregates probabilities from the initial value up to \( x \).
• For continuous distributions, the CDF is an integral of the pdf, essentially summing up area under the curve of the probability density function until \( x \).
• In the case of a Weibull distribution, the CDF \( F_W(w) = 1 - e^{-(\lambda w)^\alpha} \) highlights how likely it is for the modeled event to occur by a certain time \( w \). This function offers insights into failure times, among others.

The CDF is crucial for interpreting and visualizing a distribution, helping statisticians and engineers alike to predict behavior over time and assess risk.