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It is a weighted mixture of exponentials with different rates

It is symmetric

The geometric distribution models:

The waiting time between two events

The number of rejections in a sampling process

The number of successes in a fixed number of trials

The number of trials until the first success

What is the key advantage of the Ziggurat Method compared to simpler Accept-Reject approaches?

It guarantees 100% acceptance rate

It achieves high efficiency through precomputed regions

It doesn't require any uniform random numbers

When using the Central Limit Theorem to generate normal random variables, what happens as you increase the number of uniform variables being summed?

The method becomes less statistically valid

The variance of the generated samples increases

The computation time decreases exponentially

The approximation to normality becomes better

The Erlang distribution can be seen as:

A discrete approximation to a normal

The sum of several independent exponential random variables

The product of uniform random variables

Why might someone choose the Mimicry Method over the Inverse Transformation Method for generating binomial random variables?

The binomial CDF is difficult to invert analytically

The Mimicry Method always produces more accurate results

Mimicking the actual Bernoulli trial process is more intuitive

The Inverse Transformation Method cannot handle discrete distributions

In the Accept-Reject method, if the bounding constant c is too large, what is the primary consequence?

The acceptance rate decreases, making the method less efficient

The method becomes mathematically invalid

The method will only generate values from the tails of the distribution

The generated samples will have incorrect mean and variance

The Composition Method is most naturally applied to which type of distribution?

Multivariate normal distributions

Distributions that are explicitly defined as mixtures

Distributions with heavy tails

Distributions with simple inverse CDFs

When generating exponential random variables using the Inverse Transformation Method, why can we use x = -ln(u)/λ instead of x = -ln(1-u)/λ?

The two expressions are mathematically identical

Both u and 1-u follow the same uniform distribution

Using u directly is computationally faster

The ln(1-u) formulation is mathematically invalid

When using the inverse transform method, what property must the distribution’s CDF have for the method to work directly?

It must be invertible

It must be symmetric

It must be differentiable

It must be bounded between 0 and 2