Statistics: Random Variables

Definition and Types

• A random variable is a variable whose possible values are determined by chance, and each value has a probability associated with it.
• Discrete random variables: can take on only specific, distinct values (e.g. rolling a die, coin toss).
• Continuous random variables: can take on any value within a certain range or interval (e.g. height of a person, duration of a phone call).

Properties

• Probability distribution: a function that describes the probability of each possible value of a random variable.
• Mean (Expected Value): the average value of a random variable, denoted by μ (mu).
• Variance: a measure of the spread or dispersion of a random variable, denoted by σ² (sigma squared).
• Standard Deviation: the square root of the variance, denoted by σ (sigma).

Discrete Random Variables

• Probability mass function (PMF): a function that describes the probability of each possible value of a discrete random variable.
• Cumulative distribution function (CDF): a function that describes the probability that a discrete random variable takes on a value less than or equal to a given value.

Continuous Random Variables

• Probability density function (PDF): a function that describes the probability of each possible value of a continuous random variable.
• Cumulative distribution function (CDF): a function that describes the probability that a continuous random variable takes on a value less than or equal to a given value.

Important Distributions

• Bernoulli distribution: a discrete distribution that models a single binary outcome (e.g. coin toss).
• Binomial distribution: a discrete distribution that models the number of successes in a fixed number of independent trials (e.g. number of heads in 10 coin tosses).
• Uniform distribution: a continuous distribution that models a random variable with equal probability of taking on any value within a certain range (e.g. random number between 0 and 1).
• Normal distribution (Gaussian distribution): a continuous distribution that models a random variable with a symmetric, bell-shaped curve (e.g. height of a person, IQ score).

Definition and Types

• A random variable is a variable whose possible values are determined by chance, and each value has a probability associated with it.
• Discrete random variables can take on only specific, distinct values, such as rolling a die or coin toss.
• Continuous random variables can take on any value within a certain range or interval, such as height of a person or duration of a phone call.

Properties

• A probability distribution is a function that describes the probability of each possible value of a random variable.
• The mean (expected value) of a random variable is the average value, denoted by μ (mu).
• The variance of a random variable is a measure of the spread or dispersion, denoted by σ² (sigma squared).
• The standard deviation of a random variable is the square root of the variance, denoted by σ (sigma).

Discrete Random Variables

• A probability mass function (PMF) describes the probability of each possible value of a discrete random variable.
• A cumulative distribution function (CDF) describes the probability that a discrete random variable takes on a value less than or equal to a given value.

Continuous Random Variables

• A probability density function (PDF) describes the probability of each possible value of a continuous random variable.
• A cumulative distribution function (CDF) describes the probability that a continuous random variable takes on a value less than or equal to a given value.

Important Distributions

• The Bernoulli distribution is a discrete distribution that models a single binary outcome, such as a coin toss.
• The binomial distribution is a discrete distribution that models the number of successes in a fixed number of independent trials, such as the number of heads in 10 coin tosses.
• The uniform distribution is a continuous distribution that models a random variable with equal probability of taking on any value within a certain range, such as a random number between 0 and 1.
• The normal distribution (Gaussian distribution) is a continuous distribution that models a random variable with a symmetric, bell-shaped curve, such as height of a person or IQ score.