Discrete Random Variables

Point Mass Distribution

Definition

has a point mass distribution at , written , if in which case

The probability mass function is for , and 0 otherwise.

Discrete Uniform Distribution

https://en.wikipedia.org/wiki/Discrete_uniform_distribution

Definition

Let be a given integer. Suppose has a probability mass function given by

We then say that has a uniform distribution on .

Bernoulli Distribution

https://en.wikipedia.org/wiki/Bernoulli_distribution

Definition

Let represent a binary coin flip. Then and for some . We say that has a Bernoulli distribution written . The probability mass function is given by

for .

Generating Bernoulli Samples

template <typename T>
auto bernoulli(T p, Generator gen) -> T {
    const auto u = uniform_real_sample(0, 1, gen);
    return static_cast<T>(u < p);
}

Binomial Distribution

https://en.wikipedia.org/wiki/Binomial_distribution

Definition

The probability of getting exactly successes in independent Bernoulli trials (with the same rate ) is given by the probability mass function

for . A random variable with this mass function is called a Binomial random variable, and we write .

Theorem

If and , then .

Generating Binomial Samples

template <typename T>
auto binomial(T p, T n, Generator gen) -> T {
    T sum{};
    for (auto _ : std::views::iota(0, n)) {
        const auto u = uniform_real_sample(0, 1, gen);
        sum += static_cast<T>(u < p);
    }
}

Geometric Distribution

https://en.wikipedia.org/wiki/Geometric_distribution

Definition

The Geometric distribution is either one of the two discrete probability distributions:

• The probability distribution of the number of Bernoulli trials needed to get one success, supported on
• The probability distribution of the number of failures before the first success, supported on .

If the probability of success on each trial is , then the probability that the -th trial is the first success is

for . The above form of the geometric distribution is used for modeling the number of trials up to and including the first success. By contrast, the following form of the geometric distribution is used for modeling the number of failures until the first success

for .

Generating Geometric Samples

template <typename T>
auto geometric(T p, Generator gen) -> T {
    const auto u = uniform_real_sample(0, 1, gen);
    return p == 1 ? 1 : std::ceil(std::log1p(-u) / std::log1p(-p));
}

Poisson Distribution

https://en.wikipedia.org/wiki/Poisson_distribution

Definition

A discrete random variable is said to have a Poisson distribution with parameter , denoted as , if it has the probability mass function given by

where is the number of occurrences. The Poisson distribution is often used as a model for the counts of rare events.

Theorem

If and , then .

Generating Poisson Samples

Devroye, Luc, and Luc Devroye. “Discrete univariate distributions.” Non-Uniform Random Variate Generation (1986): 485-553.

template <typename T>
auto poisson(T lambda, Generator gen) -> T {
    T x{};
    auto p = std::exp(-lambda);
    auto s = p;
    const auto u = uniform_real_sample(static_cast<T>(0), static_cast<T>(1), gen);
    while (u > s) {
        ++x;
        p *= lambda / x;
        s += p;
    }
    return x;
}

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 2.