Random Variable Types to Know for Intro to Probabilistic Methods

Random variables are essential in understanding uncertainty in various situations. They can be discrete, taking specific values, or continuous, covering a range. Different types, like Bernoulli and Poisson, help model real-world scenarios effectively.

1. Discrete Random Variables

   • Take on a countable number of distinct values.
   • Examples include the number of students in a class or the outcome of rolling a die.
   • Probability mass function (PMF) is used to describe the probabilities of each possible value.

2. Continuous Random Variables

   • Can take on an infinite number of values within a given range.
   • Examples include height, weight, or temperature.
   • Described by a probability density function (PDF), where probabilities are found over intervals rather than specific values.

3. Bernoulli Random Variables

   • A special case of discrete random variables with only two possible outcomes: success (1) or failure (0).
   • Used to model binary outcomes, such as flipping a coin or passing a test.
   • The probability of success is denoted by p, and the probability of failure is 1 - p.

4. Binomial Random Variables

   • Represents the number of successes in a fixed number of independent Bernoulli trials.
   • Defined by two parameters: the number of trials (n) and the probability of success (p).
   • The binomial distribution is used to calculate the probability of obtaining a certain number of successes.

5. Poisson Random Variables

   • Models the number of events occurring in a fixed interval of time or space, given a known average rate (λ).
   • Useful for rare events, such as the number of phone calls received at a call center in an hour.
   • The Poisson distribution is characterized by its parameter λ, which is both the mean and variance.

6. Uniform Random Variables

   • All outcomes are equally likely within a specified range.
   • Can be discrete (e.g., rolling a fair die) or continuous (e.g., selecting a number between 0 and 1).
   • The uniform distribution is defined by its minimum and maximum values.

7. Normal (Gaussian) Random Variables

   • Characterized by a bell-shaped curve, defined by its mean (μ) and standard deviation (σ).
   • Many natural phenomena are approximately normally distributed, such as heights or test scores.
   • The central limit theorem states that the sum of a large number of independent random variables tends to be normally distributed.

8. Exponential Random Variables

   • Models the time until an event occurs, such as the time until a radioactive particle decays.
   • Defined by a single parameter (λ), which is the rate of occurrence.
   • The exponential distribution is memoryless, meaning the probability of an event occurring in the next interval is independent of how much time has already passed.

9. Geometric Random Variables

   • Represents the number of trials until the first success in a series of independent Bernoulli trials.
   • Defined by the probability of success (p) on each trial.
   • The geometric distribution is useful for modeling scenarios like the number of coin flips until the first heads appears.

10. Hypergeometric Random Variables

    • Models the number of successes in a sample drawn without replacement from a finite population.
    • Defined by the population size (N), the number of successes in the population (K), and the sample size (n).
    • The hypergeometric distribution is used in scenarios like quality control testing or lottery draws.