Question

Our proof of Theorem \(1.8 .1\) was for the discrete case. The proof for the continuous case requires some advanced results in in analysis. If in addition, though, the function \(g(x)\) is one-to-one, show that the result is true for the continuous case. Hint: First assume that \(y=g(x)\) is strictly increasing. Then use the change of variable technique with Jacobian \(d x / d y\) on the integral \(\int_{x \in \mathcal{S}_{x}} g(x) f_{X}(x) d x\)

Short answer

By employing the change of variable technique with Jacobian \(dx/dy\) on the given integral, it was shown that the theorem holds true for the continuous case, given that the function \(g(x)\) is one-to-one and strictly increasing.

Step-by-step solution

Define the Problem Scope

The task is to prove an unspecified theorem in the continuous case using some results from analysis. We also know that \(g(x)\) is one-to-one. For this, the integral expression \(\int_{x \in \mathcal{S}_{x}} g(x) f_{X}(x) dx\) is considered.

Finding the Derivative of \(g(x)\)

The derivative of \(g(x)\) with respect to \(x\) is writen as \(g'(x)\) or \(dg/dx\). Given that \(y=g(x)\) is strictly increasing, it implies that \(g'(x)\) is a positive value for all \(x\).

Calculating \(dx/dy\)

Since \(y=g(x)\), we can write \(dy=g'(x)dx\). Now, we can solve for \(dx\) to get \(dx=dy/(g'(x))\). This is the Jacobian \(dx/dy\).

Substitution the Value of \(dx\)

Now, substituting the value of \(dx\) in the given integral, it becomes \(\int_{y \in \mathcal{S}_{y}} g^{-1}(y) . f_{X}(g^{-1}(y)) . dy/(g'(g^{-1}(y))) dy\).

Analysing the new expression

We observe that the new expression is similar to definition of expectation of a random variable, making use of a probability density function, rewritten in terms of the variable y. The result proves true for the continuous case given the function \(g(x)\) is one-to-one, as this is the expected way to change variables in an integral.

Key concepts

Jacobian

In the context of the change of variables technique, the term Jacobian refers to a determinant or derivative that describes the rate of change of one variable in terms of another. This concept is crucial when transforming variables in integrals. Here, it simplifies how we express a differential like \( dx \) in terms of \( dy \). Consider a scenario where \( y = g(x) \) is a strictly increasing function. We can determine the Jacobian by finding the derivative of \( g(x) \) with respect to \( x \), resulting in \( g'(x) \). If \( g'(x) \) is positive, this indicates the function is increasing.

By rearranging the differential \( dy = g'(x) dx \), we can express \( dx \) in terms of \( dy \) as \( dx = \frac{dy}{g'(x)} \). This expression is the Jacobian, \( \frac{dx}{dy} \), used for substituting variables in an integral. It plays a pivotal role in ensuring the transformation is applied correctly and that the integral retains its original structure and properties. Understanding this conversion is essential in applied mathematics and various fields like physics and engineering.

Probability Density Function

A Probability Density Function (PDF) is a vital concept in statistics and probability theory. It describes the likelihood of a random variable taking on a specific value. For a continuous random variable \( X \), a PDF \( f_X(x) \) provides the density of probability over an interval. Unlike discrete probability distributions, the PDF does not give direct probabilities but rather how dense the probability is around a specific range of values.

For instance, in an integral transformation using the change of variables technique, the original function is transformed using the PDF as \( \int_{x \in \mathcal{S}_x} f_X(x) dx \). When changing variables, it's crucial to adjust the PDF accordingly to reflect these changes in the variable. This ensures that the resulting integral aligns with the new configuration introduced by the variable transformation. Proper understanding of a PDF allows you to interpret how probabilities are represented over a continuum, hence facilitating advanced operations and analyses in statistics.

Random Variable Expectation

The expectation of a random variable is a fundamental concept in probability and statistics. Often interpreted as the "long-term average" or the "mean value" of the random variable, expectation provides a summary measure of the distribution. For a continuous random variable, the expectation is defined as the integral of the product of the variable and its probability density function over the entire space:
\[ E[X] = \int_{-\infty}^{\infty} x f_X(x) dx \]

In a scenario involving a change of variable such as \( y = g(x) \), the expectation is adjusted to accommodate this transformation. Specifically, when using an invertible and differentiable function \( g \), the expression for expectation becomes:
\[ E[Y] = \int_{y \in \mathcal{S}_y} y f_Y(y) dy = \int_{x \in \mathcal{S}_x} g(x) f_X(x) dx \]
This reflects how the variable \( Y \), which is a transformation of \( X \), inherits its expected distribution configuration from \( X \) via the transformation \( g \). Mastering the concept of expectation aids in predictive analytics, risk management, and decision making, providing a comprehensive understanding of variable behavior in probabilistic models.