Problem 192 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose the velocity of a gas molecule of mass \(m\) is a random variable with pdf \(f_{Y}(y)=a y^{2} e^{-b y^{2}}, y \geq 0\), where \(a\) and \(b\) are positive constants depending on the gas. Find the pdf of the kinetic energy, \(W=(m / 2) Y^{2}\), of such a molecule.

Step-by-Step Solution

Answer

The probability density function of the kinetic energy of the molecule is given by \(f_{W}(w) = ae^{-2bW/m}\sqrt{\frac{2W}{m^3}}.\)

Step 1: Identify the equations

We are given that the probability density function \(f_{Y}(y) = ay^2 e^{-by^2}\) for the velocity. The kinetic energy of the molecule is given by \(W = \frac{m}{2}Y^2\).

Step 2: Solve for Y in terms of W

To enable substitution, we rewrite the kinetic energy equation as \(y = \sqrt{\frac{2W}{m}}\).

Step 3: Substitute y in pdf of Y

Substitute \(y = \sqrt{\frac{2W}{m}}\) into \(f_{Y}(y) = ay^2 e^{-by^2}\), to get \(f_{Y}(\sqrt{\frac{2W}{m}}) = a\frac{2W}{m} e^{-b(\frac{2W}{m})}\).

Step 4: Differentiate y=w.r.t W

During the transformation of the random variable, we need to find the derivative of \(y\) with respect to \(W\), i.e., \(y'=\frac{dy}{dW}=\frac{1}{\sqrt{2W/m}}\). Manuscript the integral of its distribution function.

Step 5: Calculate pdf of W

To evaluate the pdf of the new variable \(W\), use the relationship between the pdf of the transformed variable and the original variable: \(f_{W}(w) = |y'|\cdot f_Y(y=\sqrt{\frac{2W}{m}}) =|\frac{1}{\sqrt{2W/m}} |(a\frac{2W}{m} e^{-b(\frac{2W}{m})}) = ae^{-2bW/m}\sqrt{\frac{2W}{m^3}}.\).

Key Concepts

Kinetic Energy Transformation

Kinetic energy is a fundamental concept in physics, describing the energy an object possesses due to its motion. When a molecule moves, its kinetic energy can be calculated using the formula \[W = \frac{m}{2} Y^2\]where \(W\) is kinetic energy, \(m\) is mass, and \(Y\) is velocity.
This transformation expresses kinetic energy as a function of velocity, which is often needed when velocity is a random variable.

In the context of the exercise, we need to find the probability density function (PDF) of the kinetic energy \(W\) by using the given PDF of the velocity \(Y\).
We start by identifying the PDF of \(Y\) and then replacing \(Y\) in terms of \(W\).
It's crucial to remember that when transforming variables, like from velocity to kinetic energy, formulas will change whilst keeping their relationship intact. This transformation alters the domain and behavior of the associated probability density function.

Random Variable Transformation

Random variable transformation is a method used to find the PDF of a new variable that is a function of a known random variable. In this exercise, the velocity \(Y\) of a gas molecule is a random variable, meaning its value is determined by chance.
The challenge is to determine how this random variable transforms into another random variable, the kinetic energy \(W\).

To perform the transformation, we start by expressing the new variable \(W\) in terms of \(Y\) using the given relationship: \[y = \sqrt{\frac{2W}{m}}.\]Once \(y\) is expressed in terms of \(W\), we substitute it back into the original PDF of \(Y\).
Don't forget to calculate the derivative of the transformation function, known as the Jacobian, which accounts for scaling and changes in distribution.
The absolute value of this derivative is then multiplied by the transformed PDF to obtain the new PDF of \(W\), giving insight into how probability distributes over the kinetic energy of the molecule.

Probability Theory

Probability theory is the mathematical framework for quantifying uncertainty and is fundamental to understanding random variables.
A probability density function (PDF) helps specify the likelihood of a continuous random variable falling within a particular range of values.

In this exercise, we're given the PDF of the velocity \(Y\) as \(f_Y(y) = ay^2 e^{-by^2}\).
This function describes how likely it is to observe a particular velocity for a molecule.
Our task is to transform this PDF to align with the kinetic energy \(W\).

The transformation involves not just substituting variables, but also involves mathematical operations including differentiation.
After transforming, the new PDF describes how probability is distributed across different values of kinetic energy.
Using techniques from probability theory, such as understanding relationships between different random variables and calculating derivatives, allows us to derive such new probability distributions. This understanding is crucial for modeling real-world systems where variables naturally undergo transformations.