# Discrete random variables and distributions_imported Flashcards

What are discrete random variables?

Discrete random variables are variables that can take on a finite or countably infinite number of distinct values. These values are determined by the outcome of a random experiment or process.

What kind of values can discrete random variables take?

Discrete random variables can take on specific, distinct values, usually represented by integers. For example, the number of heads in a series of coin tosses or the outcome of rolling a fair six-sided die.

How are probabilities associated with discrete random variables?

Each possible value of a discrete random variable is associated with a probability of occurrence. These probabilities are often represented using a probability mass function (PMF).

What does the probability distribution of a discrete random variable show?

The probability distribution of a discrete random variable shows the probabilities of all possible outcomes. It provides insights into the likelihood of each value occurring.

Can you give an example of a discrete random variable and its probability distribution?

Certainly! Consider a random variable X representing the number of heads in two coin tosses. It can take values 0, 1, or 2. The associated probabilities might be P(X = 0) = 0.25, P(X = 1) = 0.5, and P(X = 2) = 0.25.

What is the requirement for the sum of probabilities in a discrete probability distribution?

The sum of the probabilities for all possible values of a discrete random variable must equal 1. This ensures that one of the possible outcomes will indeed occur.

Are there any measures used to describe the distribution of discrete random variables?

Yes, discrete random variables have well-defined mean (expected value) and variance, which provide measures of central tendency and spread for the variable’s distribution.

Can you provide some common examples of discrete random variables?

Certainly! Examples of discrete random variables include the outcomes of dice rolls (e.g., rolling a six-sided die), the number of emails received in an hour, the number of defects in a production batch, and the number of people in a household.

What is a Bernoulli random variable?

A Bernoulli random variable represents a binary outcome (success or failure) of a single trial.

Example: What is the probability of getting a Heads when flipping a fair coin?

The probability of getting a Heads when flipping a fair coin is 0.5.

What is a Binomial random variable?

A Binomial random variable represents the number of successes in a fixed number of independent Bernoulli trials.

Example: What is the probability of getting exactly 3 Heads in 5 coin tosses?

The probability of getting exactly 3 Heads in 5 coin tosses is given by the binomial distribution formula.

What is a Poisson random variable?

A Poisson random variable represents the number of events occurring in a fixed interval of time or space.

Example: What is the probability of observing 2 car accidents at a specific intersection in a given hour?

The probability of observing 2 car accidents in a given hour can be calculated using the Poisson distribution formula.

What is a Geometric random variable?

A Geometric random variable represents the number of trials needed for the first success in a sequence of Bernoulli trials.

Example: How many times do you need to roll a die to get the first 6?

The expected number of rolls needed to get the first 6 on a fair 6-sided die is 6.

What is a Hypergeometric random variable?

A Hypergeometric random variable represents the number of successes in a sample drawn without replacement from a finite population.

Example: If you draw 2 cards from a deck without replacement, what’s the probability of getting exactly 1 Ace?

The probability of drawing exactly 1 Ace from a deck when drawing 2 cards without replacement can be calculated using the hypergeometric distribution formula.