12.1 Probability Distributions and Random Variables

Probability distributions are the backbone of statistical physics, helping us understand random events in the physical world. They quantify the likelihood of different outcomes for variables like particle positions or energy states, giving us a mathematical framework to analyze complex systems.

From quantum mechanics to error analysis, these distributions play a crucial role in various areas of physics. By calculating moments like expectation values and variances, we can extract meaningful information about physical observables and make predictions about system behavior.

Probability Distributions

Probability distributions in physics

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• Probability distributions quantify the likelihood of different outcomes for a random variable in a physical system
  • Discrete random variables have a countable number of possible values (number of particles in a quantum state)
  • Continuous random variables can take on any value within a specified range (position of a particle in a potential well)
• Probability mass function (PMF) describes the probability distribution for discrete random variables
  • Denoted as $P(X = x)$, where $X$ is the random variable and $x$ is a specific value
  • Must satisfy the conditions $0 \leq P(X = x) \leq 1$ and the sum of all probabilities equals 1, $\sum_x P(X = x) = 1$
• Probability density function (PDF) describes the probability distribution for continuous random variables
  • Denoted as $f(x)$, where $x$ is a specific value of the random variable
  • Must satisfy the conditions $f(x) \geq 0$ and the integral of the PDF over the entire domain equals 1, $\int_{-\infty}^{\infty} f(x) dx = 1$
• Cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a specific value
  • For discrete random variables, the CDF is the sum of probabilities up to a given value: $F(x) = P(X \leq x) = \sum_{t \leq x} P(X = t)$
  • For continuous random variables, the CDF is the integral of the PDF up to a given value: $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$

Moments of probability distributions

• Expectation value (mean) of a random variable $X$ represents the average value of the variable
  • For discrete random variables, the expectation value is the sum of each value multiplied by its probability: $E[X] = \sum_x x P(X = x)$
  • For continuous random variables, the expectation value is the integral of each value multiplied by its probability density: $E[X] = \int_{-\infty}^{\infty} x f(x) dx$
• Variance of a random variable $X$ measures the spread of the distribution around the mean
  • Calculated as the expectation value of the squared deviation from the mean: $Var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2$
• Standard deviation is the square root of the variance: $\sigma = \sqrt{Var(X)}$
• Higher moments of a distribution characterize its shape and properties
  • $n$-th moment is calculated as $E[X^n] = \sum_x x^n P(X = x)$ for discrete random variables and $E[X^n] = \int_{-\infty}^{\infty} x^n f(x) dx$ for continuous random variables
  • Skewness (3rd moment) measures the asymmetry of the distribution
  • Kurtosis (4th moment) measures the heaviness of the tails of the distribution compared to a normal distribution

Applications of probability theory

• Quantum mechanics: The wave function $\psi(x)$ is a probability amplitude, and the square of its absolute value, $|\psi(x)|^2$, represents the probability density for finding a particle at position $x$
• Statistical mechanics: Probability distributions such as Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distributions describe the likelihood of a particle occupying a specific energy state in a system at thermal equilibrium
• Error analysis: Propagation of uncertainties in measurements can be analyzed using probability distributions to determine the likelihood of different outcomes and the overall uncertainty in the final result

Distributions and physical observables

• Expectation values of physical observables correspond to the mean of the associated probability distribution
  • The average energy of a system is the expectation value of the energy distribution, $\langle E \rangle = \int E P(E) dE$
• Variances of physical observables relate to the spread of the associated probability distribution
  • The uncertainty principle in quantum mechanics, $\Delta x \Delta p \geq \hbar/2$, connects the uncertainties in position and momentum to the spread of their respective probability distributions
• Moments of probability distributions can be used to characterize physical properties
  • Dipole moment (1st moment), quadrupole moment (2nd moment), and higher-order moments describe the charge or mass distribution in electrostatics and magnetostatics