The speed of a molecule in a uniform gas at equilibrium is a...

The speed of a molecule in a uniform gas at equilibrium is a random variable $V$ whose probability distribution is given by $$f(v)=\left\{\begin{array}{ll}k v^{2} e^{-b v^{2}}, & v>0 \\0, & \text { elsewhere }\end{array}\right.$$ where $k$ is an appropriate constant and $b$ depends on the absolute temperature and mass of the molecule. Find the probability distribution of the kinetic energy of the molecule $W,$ where $W=m V^{2} / 2$.

The speed of a molecule in a uniform gas at equilibrium is a random variable $V$ whose probability distribution is given by
$$f(v)=\left\{\begin{array}{ll}k v^{2} e^{-b v^{2}}, & v>0 \\0, & \text { elsewhere }\end{array}\right.$$
where $k$ is an appropriate constant and $b$ depends on the absolute temperature and mass of the molecule. Find the probability distribution of the kinetic energy of the molecule $W,$ where $W=m V^{2} / 2$.

Key Concepts

Transformation of Random Variables

This concept involves finding the probability distribution of a function of another random variable. It is typically done using the change-of-variable method, where one finds the inverse function and its derivative (the Jacobian) to adjust the original probability density function appropriately.

Jacobian (Change-of-Variable Technique)

When transforming random variables, the Jacobian represents the absolute value of the derivative of the inverse transformation. This technique ensures that the new probability density function, expressed in terms of the transformed variable, is correctly normalized and accounts for the stretching or compressing of the distribution.

Probability Density Functions

Probability density functions (pdfs) describe the likelihood of a continuous random variable taking on certain values. Understanding how to manipulate and transform pdfs is essential for deriving the distributions of related physical quantities or functions of the original variables.

Kinetic Energy Distribution

In physical contexts, quantities like kinetic energy are functions of other random variables, such as molecular speed. Deriving the kinetic energy distribution involves applying the transformation of random variables concept, where the functional relationship (e.g., energy being proportional to the square of speed) is taken into account to find the new pdf.

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