Problem 92 - An Introduction to Mathematical Statistics and Its Applications

Question

Demonstrate that \(\lambda\) plays the role of a scale parameter by showing that if \(Y\) is gamma with parameters \(r\) and \(\lambda\), then \(\lambda Y\) is gamma with parameters \(r\) and \(1 .\)

Step-by-Step Solution

Answer

By substituting \(y = \lambda Y\) into the probability density function of the gamma distribution and simplifying, it can be clearly demonstrated that a scale transformation by \(\lambda\) of a gamma-distributed random variable \(Y\) with parameters \(r\) and \(\lambda\) transforms it into a gamma-distributed random variable with parameters \(r\) and \(1\).

Step 1: Write down the probability density function of gamma distribution

The probability density function of a gamma distribution is given by \(f_Y(y) = \frac{y^{r-1}e^{-y/\lambda}}{\lambda^r\Gamma(r)}\) where \(y > 0\), \(r > 0\), and \(\lambda > 0\).

Step 2: Substitute λY for y in the density function

In this step, substitute \(y = \lambda Y\) into the density function to get \(f_Y(\lambda Y) = \frac{(\lambda Y)^{r-1}e^{-\lambda Y/\lambda}}{\lambda^r\Gamma(r)}\) which simplifies to \(f_Y(\lambda Y) = \frac{\lambda^{r-1}Y^{r-1}e^{-Y}}{\lambda^r\Gamma(r)}\).

Step 3: Simplify the equation

By simplifying, we get: \(f_Y(\lambda Y) = \frac{Y^{r-1}e^{-Y}}{\Gamma(r)}\) because the \(\lambda^{r-1}\) and \(\lambda^r\) will cancel out each other.

Step 4: Show the new distribution

The term derived above is the probability density function of a gamma distribution with parameters r and 1. This clearly shows that \(\lambda Y\) has a gamma distribution with parameters r and 1.

Key Concepts

Scale Parameter

The scale parameter in statistical distributions, like the gamma distribution, plays an essential role in shaping the spread of data. In the context of the gamma distribution, the scale parameter, denoted as \( \lambda \), directly influences the stretch of the distribution along the horizontal axis. To understand its impact, consider the probability density function of the gamma distribution, \( f_Y(y) = \frac{y^{r-1}e^{-y/\lambda}}{\lambda^r\Gamma(r)} \), where \( \lambda \), \( r \), and \(\(y\)\) are positive values.

When you multiply \( Y \) by \( \lambda \) — that is, consider the random variable \( \lambda Y \) — and carry out a substitution in the density function, you end up demonstrating that \( \lambda \) acts as a scale factor. As shown in the textbook solution, the substitution simplifies to a form resembling the original density function but scaled differently, which confirms that \( \lambda Y \) remains a gamma distribution with the same shape parameter \( r \) but now with a scale parameter equal to 1. This exposes the scale parameter's role in altering the distribution's spread without affecting its general shape.

Probability Density Function

The probability density function (PDF) is a mathematical expression that describes the likelihood of a random variable taking on a specific value. In simpler terms, it gives you the 'height' of the distribution at any given point, which correlates to the probability of the variable being near that point. For continuous variables like those represented by the gamma distribution, the PDF tells us how likely it is to find the random variable within a small interval around a value.

For example, the gamma distribution's PDF is \( f_Y(y) = \frac{y^{r-1}e^{-y/\lambda}}{\lambda^r\Gamma(r)} \) and reveals crucial information about the behavior of the system or process described by the distribution. By integrating the PDF over a range, we get the probability that the random variable falls within that range. The steps outlined in the exercise solution effectively deconstruct the PDF of a gamma distribution to show how scale changes when you multiply the variable \( Y \) by the scale parameter \( \lambda \) — a valuable insight for statisticians and analysts alike.

Statistical Distributions

Statistical distributions are fundamental tools in statistical analysis, depicting the values a variable can take and how often it takes them. Different types of distributions are suited to modeling different types of variables and data behaviors. The gamma distribution is a two-parameter family of continuous probability distributions that is commonly used in fields such as insurance, finance, and reliability engineering.

It is particularly useful for modeling the time until an event occurs, such as the lifespan of a piece of machinery or the time between customer arrivals. The two parameters, shape \( r \) and scale \( \lambda \) work together to define the specific characteristics of the gamma distribution in question. While the shape parameter \( r \) affects the distribution's asymmetry or 'skewness', the scale parameter \( \lambda \) adjusts the concentration or dispersion of data across the range of possible values. By examining the roles of these parameters, students and professionals can gain a better grasp of how the gamma distribution can model complex, real-world phenomena.