Statistics: Random Variables and Models

Models, Random Variables, and Realizations

• Models are simplified representations of real-world phenomena. Their quality is assessed by comparing predictions to observed reality.

Review of Random Variables

• A random variable is a numerical outcome of an experiment yet to be performed.
• A discrete random variable has a finite or countable number of possible values.
• A continuous random variable has an uncountably infinite number of possible values.

Discrete Random Variable Example

• An example experiment is shooting two basketball free throws.
• The random variable, Y, is the number of successful shots.
• Possible values are {0, 1, 2}.
• The probability mass function (pmf) specifies the probability of each outcome.

Expected Value and Variance

• The expected value, E(X), is the mean of a probability distribution.
• E(X) is calculated by summing the product of each value and its probability for discrete variables or by integrating the product of each value and its probability density for continuous variables.
• The variance, var(X), measures the spread of a distribution. It is calculated by taking the expected value of the square of the difference between X and its expected value.

Independent and Identically Distributed (i.i.d.) Random Variables

• Random variables are independent if the outcome of one variable doesn't affect the outcome of another.
• Random variables are identically distributed if they all share the same probability distribution.

Parameters

• Parameters describe the probability distribution.
• Examples include the mean, standard deviation, etc.
• Often represented using Greek letters (e.g., μ, σ).

The Uniform Distribution

• A continuous random variable X has a uniform distribution between a and b (Unif(a, b)) if its probability density function is constant within that interval.
• E(X) = (a + b)/2
• var(X) = (b - a)²/12

The Bernoulli Distribution

• A discrete random variable X follows a Bernoulli distribution with parameter θ if it represents a single Bernoulli trial (success/failure).
• X is 1 if the outcome is a success, 0 for failure.
• The probability of success is θ.
• E(X) = θ
• var(X) = θ(1 - θ)

The Binomial Distribution

• A discrete random variable X follows a binomial distribution with parameters n and θ if it represents the number of successful trials in n independent Bernoulli trials.
• E(X) = nθ
• var(X) = nθ(1 - θ)

The Normal Distribution

• A continuous random variable X has a Normal distribution with parameters μ and σ² (N(μ, σ²)) if its probability density function (pdf) is given by a specific formula.
• E(X) = μ
• var(X) = σ²

Estimators and Estimates

• An estimator is a function that maps data to an estimate.
• An estimate is a calculation based on the data.
• Examples include the sample mean (an estimator for the expected value) for a probability distribution.
• Sample Variance (estimator for variance)

Explore the concepts of models and random variables in this quiz. Learn about discrete and continuous random variables, as well as their expected values and variances. Test your understanding of probability mass functions and real-world applications.