Random Number Generation

Extracting Random Numbers from Nature

1. State two natural sources of random numbers.
2. State what devices are used to “extract” the random numbers from the two sources in the previous learning objective.

Pseudorandom Number Generators (PRNG)

1. Explain how pseudorandom numbers differ from truly random numbers.

Linear Congruential Generators

1. State the algorithm for generating pseudorandom numbers using a linear congruential generator (LCG).
2. Given the modulus, multiplier, and increment for a LCG and a seed, generate variates from the LCG.
3. Evaluate expressions \(a \operatorname{mod} m\) involving the modulo operation by-hand.
4. Compute expressions \(a \operatorname{mod} m\) using R.
5. State the possible values that a LCG with a given modulus could generate.
6. Given iterates from a LCG with a relatively small period, identify the period of the LCG.

Desirable Properties of Variates from a PRNG

1. State three desirable properties of iterates from a PRNG.

Generating Non-uniform Variates from Uniform Variates

1. Convert variates uniform on \(\{0, 1, 2, \ldots, m - 1\}\) to variates approximately uniform on \((0, 1)\).
2. Compute the quantile function of a given distribution using R.
3. Given the quantile function \(q_{F_{X}}\) of a distribution \(F_{X}\) and variates uniform on \((0, 1)\), generate non-uniform variates following \(F_{X}\).

R’s PRNGs

1. Use R to generate pseudorandom numbers following a given distribution.
2. Set the seed of the PRNG in R, and explain what setting the seed does.
3. Explain why it may or may not be appropriate to set the seed in R.