⭐ Central Limit Theorem (CLT) The Central Limit Theorem says something surprisingly powerful: If you take many random samples and compute their means,the distribution of those sample means will be approximately normal,even if the original…
⭐ Z‑Score A z‑score tells you how many standard deviations an observation is from the mean. What it does Example Population mean , standard deviation .What is the z‑score of ? Interpretation:The value is 1.5…
⭐ Histogram vs. Density Plot Both visualize distributions, but they answer slightly different questions and behave differently. 📊 Histogram A histogram groups data into bins and shows counts (or proportions) in each bin. Key features…
A random variable that follows a geometric distribution satisfies: This means: The probability you still have to wait more trials does NOT depend on how long you’ve already been waiting. Your past failures don’t change…
The geometric distribution models the number of trials needed until the first success occurs in a sequence of independent Bernoulli trials (like repeated coin flips). Think of it as the math of “How long until…
⭐ Binomial Distribution The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. Think of it as the math of “How…
The expected value (mean) of random variables adds even if the variables are dependent. This is the magic part: Expectation is always linear — no independence required. Formally, for any random variables and : And…
A histogram is a graph that shows how data are distributed by grouping values into bins (intervals) and showing how many observations fall into each bin. It’s perfect for visualizing: Think of it as stacking…
A probability density function describes the distribution of a continuous random variable. If is continuous, its PDF is a function such that: The key idea For continuous variables: The PDF is not a probability. Probability…
A probability mass function is a function that gives the probability of each individual value of a discrete random variable. If is a discrete random variable, then its PMF is: It tells you: A PMF…
Most random variables fall into two big categories: Everything else is a refinement of these two. 🎯 1. Discrete Random Variables A discrete random variable takes countable values — usually integers. Key features Examples Common…
In a magical kingdom, training pigs to fly with brooms results in a 15% success chance; expected flying pigs are 4.5, with varied probabilities.
Quizzes In an economic context, a Bernoulli distribution can be used to model:a. The distribution of income across a populationb. The probability of a consumer making a purchase or notc. The stock market index changesd.…
