Question

Suppose \(X\) is a random variable with the pdf \(f_{X}(x)=b^{-1} f((x-a) / b)\), where \(b>0\). Suppose we can generate observations from \(f(z)\). Explain how we can generate observations from \(f_{X}(x)\).

Short answer

Firstly, generate an observation \(Z\) from \(f(z)\). Then, transform that observation using the equation \(x = a + b \cdot Z\) to obtain an observation from \(f_{X}(x)\).

Step-by-step solution

Understand the Transformation of the Random Variable

The given pdf for the random variable \(X\) is provided as \(f_{X}(x)=b^{-1} f((x-a) / b)\). This indicates that the random variable \(X\) is transformed from another random variable \(Z\) with the density function \(f(z)\). The transformation involves a scaling operation by the factor \(b\) and a translation operation by the value \(a\).

Generating Observations from \(f(z)\)

Since we know how to generate observations from the density function \(f(z)\), we can make use of this capability. Let's denote these generated observations as \(Z\).

Generating Observations from \(f_{X}(x)\)

Given the transformation of the random variable \(X\) from \(Z\), we can generate observations from \(f_{X}(x)\) using the same transformation. Expressly, an observation \(x\) from \(f_{X}(x)\) can be obtained by the equation \(x = a + b \cdot Z\). Here, \(b\) is the scaling factor and \(a\) is the translation factor. This is a standard method of obtaining samples from one distribution using samples from another distribution by applying an appropriate transformation.

Key concepts

Probability Density Function (PDF)

Understanding the probability density function (PDF) is crucial when dealing with continuous random variables. The PDF, denoted as \( f_X(x) \), represents the likelihood of a random variable taking on a particular value within a continuous range. It's the tool that helps us describe the distribution of continuous outcomes, and its integral is used to find probabilities for intervals of values. For example, the probability that a random variable \( X \) falls between two points \( a \) and \( b \) is calculated through the integral of its PDF from \( a \) to \( b \).

In the given exercise, \( X \) is a random variable whose PDF has been transformed by scaling and translation parameters \( b \) and \( a \). The scaling modifies the spread, or variance, of the distribution, while the translation shifts the mean. To ensure that the transformed PDF is still valid, it must integrate to one over all possible values of \( X \), which is why we compensate for the scaling with the factor of \( b^{-1} \). Translation does not affect the total probability, so the PDF remains valid without additional adjustments.

Generating Random Observations

Generating random observations is a form of simulation that allows us to model and study complex systems. In statistical analysis, generating observations from a known PDF enables researchers to predict and analyze outcomes for various scenarios. These simulated observations can be drawn using different methods, such as the inverse transform method, rejection sampling, or other algorithmic techniques depending on the complexity and nature of the PDF.

In the context of our exercise, once you have generated observations \( Z \) from the base distribution with PDF \( f(z) \), you transform those observations using the given scaling factor \( b \) and translation \( a \). This approach is particularly useful when the base distribution is standard and easy to sample from, such as the normal, uniform, or exponential distributions.

Scaling and Translation of Random Variables

Scaling and translation are two fundamental concepts in transforming random variables. When we scale a random variable, we're stretching or compressing its distribution, which affects measures like the variance. For instance, multiplying a random variable by a constant \( b > 0 \) expands the distribution if \( b > 1 \), and compresses it if \( b < 1 \).

Translation involves shifting the entire distribution along the horizontal axis. Adding a constant \( a \) to a random variable shifts the distribution right if \( a > 0 \), and left if \( a < 0 \). The mean of the distribution correspondingly adjusts, but the shape of the distribution remains unchanged.

In the exercise, we apply both of these transformations to the random variable \( Z \) to generate a new variable \( X \). Through this method, we can obtain a transformed variable \( X \) with a specific mean and variance desired for our study or simulation, simply by generating base random observations \( Z \) and applying the formula \( X = a + bZ \). This practical application of transformation allows statisticians and data scientists to simulate and analyze data behavior under various conditions.