Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

The gamma probability density function is $$ f(x)= \begin{cases}C x^{\alpha-1} e^{-\beta x}, & \text { if } x>0 \\ 0, & \text { if } x \leq 0\end{cases} $$ where \(\alpha\) and \(\beta\) are positive constants. (Both the gamma and the Weibull distributions are used to model lifetimes of people, animals, and equipment.) (a) Find the value of \(C\), depending on both \(\alpha\) and \(\beta\), that makes \(f(x)\) a probability density function. (b) For the value of \(C\) found in part (a), find the value of the mean \(\mu\). (c) For the value of \(C\) found in part (a), find the variance \(\sigma^{2}\).

Step-by-Step Solution

Answer

(a) \( C = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \); (b) \( \mu = \frac{\alpha}{\beta} \); (c) \( \sigma^2 = \frac{\alpha}{\beta^2} \).

Step 1: Understanding the Probability Density Function

For a function to qualify as a probability density function (PDF), the integral of the function over its entire range must equal 1. Thus, for the gamma distribution PDF, we need \( \int_0^\infty C x^{\alpha-1} e^{-\beta x} \, dx = 1 \).

Step 2: Solving for Constant C

Integrate \( \int_0^\infty x^{\alpha-1} e^{-\beta x} \, dx \) using the substitution \( u = \beta x \), which gives \( du = \beta \, dx \) or \( dx = \frac{du}{\beta} \). Replace \( x \) with \( \frac{u}{\beta} \) in the integral: \( \int_0^\infty \left(\frac{u}{\beta}\right)^{\alpha-1} e^{-u} \frac{du}{\beta} \). This simplifies to \( \frac{1}{\beta^{\alpha}} \int_0^\infty u^{\alpha-1} e^{-u} \, du \), which is equal to \( \frac{1}{\beta^{\alpha}} \Gamma(\alpha) \), where \( \Gamma(\alpha) \) is the gamma function. Set this equal to \( \frac{1}{C} \) and solve for \( C \): \( C = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \).

Step 3: Calculating the Mean (\( \mu \))

The mean of a gamma distribution is given by \( \mu = \frac{\alpha}{\beta} \). Substitute the values of \( \alpha \) and \( \beta \) to find the mean.

Step 4: Calculating the Variance (\( \sigma^2 \))

The variance of a gamma distribution is given by \( \sigma^2 = \frac{\alpha}{\beta^2} \). Use the known values of \( \alpha \) and \( \beta \) to compute the variance.

Key Concepts

Probability Density Function

A probability density function (PDF) helps us understand how probabilities are distributed over a set of continuous values. For any function to be considered a PDF, the total area under its curve must be exactly 1. This ensures that the probability across all possible outcomes sums up to 100%. In the context of the gamma distribution, the PDF is defined as follows: \[ f(x)= \begin{cases}C x^{\alpha-1} e^{-\beta x}, & x>0 \ 0, & x \leq 0\end{cases} \] where \( C \) is a constant that makes sure the total area under the curve equals 1. This function describes how the probabilities are assigned over the positive real line, ensuring the distribution respects the property of a PDF.

Gamma Function

The gamma function, denoted by \( \Gamma(\alpha) \), is a crucial part of the gamma distribution. It helps in solving for the constant \( C \) in the probability density function. Mathematically, the gamma function is defined as: \[ \Gamma(\alpha) = \int_0^\infty t^{\alpha-1} e^{-t} \, dt \] You can think of the gamma function as an extension of the factorial function for real and complex numbers. While the factorial function \( n! \) exists only for non-negative integers, the gamma function fills in the gaps for non-integers. This is essential for continuous probability distributions like the gamma distribution, where \( \alpha \) can be any positive real number. Using \( \Gamma(\alpha) \), we can compute the exact expression for the normalization constant \( C \) in the PDF.

Mean and Variance

Mean and variance are pivotal in describing the shape and spread of a distribution. For the gamma distribution, the mean \( \mu \) is the average value around which the data points tend to cluster. It is given by the formula: \[ \mu = \frac{\alpha}{\beta} \] The variance \( \sigma^2 \), on the other hand, measures the spread of the distribution, indicating how much the values deviate from the mean. It is calculated as: \[ \sigma^2 = \frac{\alpha}{\beta^2} \] Both these parameters depend on \( \alpha \) and \( \beta \), which shape the gamma distribution to model various types of data effectively. Adjusting these parameters allows the gamma distribution to fit different kinds of data, making it versatile in statistical modeling.

Lifetime Distribution Models

Lifetime distribution models are used to predict the lifespan of objects, individuals, or systems. The gamma distribution is one of the commonly used models for such predictions. It helps in understanding and modeling time-to-failure or survival data. - **Versatility:** The gamma distribution's ability to shape data according to parameter variations makes it a preferred choice. - **Applications:** It finds applications in fields like finance, engineering, and biology to model lifetimes. - **Use Cases:** For example, it can model the time until an electronic component fails or the waiting time until the next seismic event occurs. The flexibility and relevance of the gamma distribution in these practical scenarios make it an invaluable tool in predictive analytics and risk assessment.