Student’s t-test & Student’s t-distribution
A t‑test is a hypothesis test used when you want to compare means but you don’t know the population standard deviation and your sample size is small. It’s used for: One‑sample t‑test → compare one…
The choice of significance level in hypothesis testing
The significance level, usually written as , is the threshold for how much evidence you require before rejecting the null hypothesis. It is the probability of making a Type I error: So choosing α is…
hypothesis testing using p-value
The p‑value is the probability of getting a result as extreme as (or more extreme than) your sample result if the null hypothesis were true. In other words: The p‑value tells you how surprising your…
hypothesis testing with critical values
Instead of using a p‑value, you compare your test statistic (like a z‑score or t‑score) to a critical value that marks the boundary of the rejection region. If your test statistic falls beyond the critical…
Fisher vs Neyman battle
🧪 Ronald Fisher: The p‑value Rebel Philosophy: Evidence, not decisions Fisher believed statistics should help scientists measure evidence against a null hypothesis. Key ideas Fisher’s vibe: The scientist as a detective, gathering clues and weighing…
setting up the hypothesis
At the heart of every hypothesis test are two competing statements about a population: They must be:Mutually exclusive (can’t both be true)Exhaustive (cover all possibilities)About population parameters, not sample statistics Let’s break down how to…
What’s hypothesis testing
Hypothesis testing is a structured way to use sample data to make decisions or draw conclusions about a population. It answers questions like: It’s the backbone of inferential statistics. 🎯 The Core Idea You start…
central limit theorem & confident interval
⭐ 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…
Normalization & z-score
⭐ 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 versus density
⭐ 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…
geometric distribution is memoryless
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…
geometric distribution
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
⭐ 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…
expectation is linear
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…
histogram
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…
probability density
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…
probability mass function
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…
Types of random variables
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…
independent vs mutually exclusive
Independent vs. Mutually Exclusive 🎯 Mutually Exclusive (Disjoint) Events Two events are mutually exclusive if they cannot happen at the same time. Example:Rolling a die: 🎯 Independent Events Two events are independent if knowing one…
general addition rule
⭐ General Addition Rule The general addition rule tells you how to find the probability that A or B happens — even when the events overlap. 📌 The formula Why subtract the intersection?Because if A…