Probability I: Distribution Functions

A function that describes the probability that a random variable takes on a value less than or equal to a given value.

A non-decreasing function that takes on values between 0 and 1. It reflects the likelihood of a random variable being less than or equal to a given value.

It describes the probability distribution of a discrete random variable. It gives the probability of a specific value.

It describes the probability distribution of a continuous random variable. It gives the probability density at a specific point.

A single outcome with two possible results: success (1) or failure (0).

A random variable associated with a Bernoulli trial, taking on a value of 1 for success and 0 for failure.

A distribution where the probability of success (p) remains constant across independent trials.

A distribution that represents the probability of getting 'k' successes in 'n' independent Bernoulli trials, where each trial has the same probability of success.

A random variable that represents the outcome of a single trial with two possible results, often called 'success' or 'failure'.

The probability of a Bernoulli random variable taking the value 1 (success).

The probability of a Bernoulli random variable taking the value 0 (failure).

The expected value or average outcome of a Bernoulli random variable.

The measure of how spread out the values of a Bernoulli random variable are from the expected value.

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

The probability of getting exactly x successes in n trials of a Bernoulli random variable with probability of success p.

The conditions required for a random variable to have a binomial distribution.

The probability of observing exactly x successes in n trials, where p is the probability of success in a single trial.

The expected value or mean of the binomial random variable X, calculated as n * p, where n is the number of trials and p is the probability of success in each trial.

A measure of how spread out the binomial distribution is, calculated as n * p * (1 - p), where n is the number of trials and p is the probability of success in each trial.

The square root of the variance, representing the average distance of outcomes from the expected value.

A random variable that quantifies the number of successes in a sample drawn from a finite population, where the probability of success is not independent across draws.

The probability of observing x successes in a sample of n items, drawn without replacement from a population with N items and k successes.

The distribution of the hypergeometric random variable.

The average value of a random variable X in a Poisson distribution. It is represented by the Greek letter lambda (λ).

The variance of a Poisson variable X is also equal to its expected value (λ).

The standard deviation of a Poisson variable X is the square root of its variance (λ).

The probability of at least two occurrences of a rare event in a large sample. It is calculated using the Poisson distribution.

A Poisson variable has parameters n (number of trials) and p (probability of success). When n is large and p is small, it can be approximated by a Poisson variable with parameter λ = np.

A method for analyzing and visualizing data to understand its structure and relationships. Techniques like histograms, scatter plots, and box plots help identify patterns and outliers.

A normal random variable with mean μ and variance σ², represented by X ~ N(μ, σ²).

A formula used to calculate the z-score for a given value of X in a normal distribution, helping to standardize and compare values across different distributions.

A statistical experiment where the probability of an event depends on the number of successful outcomes already observed. This is useful for situations like drawing cards without replacement where each event depends on the previous ones.

The average value of the hypergeometric random variable. It can be calculated by: (n * k) / N, where n is the sample size, k is the number of successes in the population, and N is the population size.

The measure of how spread out the values of a hypergeometric random variable are. Calculated using the formula: (n * k * (N-k) * (N-n)) / (N^2 * (N-1)).

The square root of the variance, indicating how much the random variable deviates from its expected value.

An event where the probability of occurrence is very small. This is commonly observed in large populations where each event is relatively rare.

A statistical distribution that models the probability of a certain number of events occurring in a fixed interval of time or space, given that these events occur independently and with a constant average rate.