Monte carlo 2

A function that maps random variables to their cumulative probabilities.

The process of finding the inverse of the cumulative probability function.

A method to generate random numbers according to a specific probability distribution using the inverse of the CDF.

A technique to generate random numbers according to a distribution when the inverse CDF method is impractical. It involves scaling the probability density function and using random numbers for acceptance or rejection.

The ratio of accepted random number pairs to the total number of pairs in the Rejection Method.

A technique to generate random numbers based on a combination of the direct method (inverse CDF) and rejection method, used when the distribution is difficult to integrate or has spikes.

The maximum value of the scaled probability density function in the Rejection Method.

The process of scaling the probability density function by its maximum value to obtain a new distribution with a maximum value of 1.

The range of values that the random variables can take in the Rejection Method.

A random number uniformly distributed between 0 and 1 used to generate random numbers in the Rejection Method.

A sequence of random numbers should be statistically independent from one another; no predictable pattern should exist.

Ideally, a random number generator should produce a sequence that never repeats; in practice, the repetition should occur after generating a very large number of values.

A sequence of random numbers should be evenly distributed within a specific range, without biases towards any particular values.

The ability to reproduce the same sequence of random numbers by setting a specific seed value, allowing for debugging and result verification across different runs.

The speed at which a random number generator produces its sequence, crucial for large-scale simulations.

The ability of a random number generator to efficiently utilize the parallel processing capabilities of modern computer architectures.

A statistical test used to assess the uniformity of a random number generator, examining the distribution of generated values to ensure they follow a specific pattern (e.g., uniform distribution).

A technique for using random numbers to represent probability distributions, enabling the simulation of real-world phenomena.

A direct method for generating samples from a probability distribution by inverting its cumulative distribution function (CDF), generating random numbers and mapping them to the corresponding values in the distribution.

A function that quantifies the likelihood of observing a specific value for a variable. For example, p(x) could represent the probability of a photon interacting at position x.

A function that represents the cumulative probability of observing a value less than or equal to a given point. It's essentially the area under the PDF curve up to that point.

The probability distribution function must always be greater than or equal to zero because negative probabilities are not possible.

The area under the probability distribution function over its entire range of possible values must equal 1. This implies that the total probability of all possible outcomes must be 100%.

A method for generating random numbers. While they are called 'random', they are actually generated using deterministic algorithms, making them 'pseudo-random'.

The method that underlies Monte Carlo simulations. It involves simulating random events based on probability functions to obtain statistical estimates of quantities of interest.

A process that transforms probability distributions into samples that represent the underlying distribution. This is crucial in Monte Carlo simulations because we need to generate random values that follow a specific distribution.

The process of modeling how photons interact (like scattering, absorption) with matter as they travel through a medium. This forms the core of Monte Carlo simulations in radiation transport.

The application of different techniques that aim to reduce the variance (errors) of Monte Carlo simulation results. This is important because reducing the variance can increase the accuracy of the simulation.

The 'heartbeat' of a Monte Carlo simulation, the Pseudo-Random Number Generator (RNG) is crucial for generating numbers that appear random. Using RNG's requires careful considerations due to their deterministic nature.