Series
Basic Statistics I Studied
13 posts
- #1
Moment Generating Function
Turns out mean and variance are just the 1st and 2nd moments — and the MGF is the tool mathematicians built to pull any moment out, cleanly, anytime!
· 3 min read - #1
Introduction
A casual dive into why sample variance divides by n-1 and how we use samples to estimate population parameters without measuring absolutely everyone.
· 4 min read - #2
Derivation of the Poisson Distribution
We kick off the distributions series by deriving the Poisson from scratch — starting with the binomial and cranking n to infinity to see what shakes out.
· 3 min read - #3
Derivation of the Exponential Distribution
We skip the normal distribution to derive the exponential distribution — aka the waiting time until the first event — straight from the Poisson!
· 5 min read - #4
Derivation of the Gamma Distribution
Deriving the Gamma distribution from scratch by connecting it to the Poisson process — turns out it's just waiting for the α-th event instead of the first!!!!
· 7 min read - #5
Derivation of the Chi-Squared Distribution
Chi-squared is just a gamma in disguise — we prove Z² follows it with 1 degree of freedom and show how sample variance ties in before jumping to the t-distribution.
· 4 min read - #6
Derivation of the Student's t-Distribution
Where the 'Student' name came from, why we ditch the Z-stat when σ is unknown, and a full derivation of the t-distribution PDF — plus properties and a worked example.
· 4 min read - #7
Derivation of the F-Distribution
We derive the F-distribution PDF from scratch — turns out it's basically next of kin to the t-distribution — then wrap up with how to actually read an F-table.
· 3 min read - #8
Hypothesis Testing
A casual walkthrough of hypothesis testing basics — null vs. alternative hypotheses, Z-statistics, and how to decide when to reject what the data's telling you.
· 9 min read - #9
A Hasty Conclusion
Hastily wrapping up the stats fundamentals series because the math foundation got too shaky — no promises on when the next part's coming, lol.
· 1 min read - #11
Gauss–Markov Theorem and the Proof of BLUE
Digging into what OLS actually guarantees — the 5 assumptions behind linear regression and why they make your estimates BLUE according to the Gauss-Markov theorem.
· 6 min read - #12
Python: Poisson, Exponential, and Gamma Distributions
Just plotted the Poisson, Exponential, and Gamma distributions in Python straight from Wikipedia's parameters — skipped the random-variable cross-checks and kept it chill this time.
· 2 min read - #13
Python: Chi-Squared, Student's t, and F Distributions
Wrapped up derivations of the chi-squared, t-, and F-distributions — and honestly it's just reminded me how shaky my stats fundamentals still are lol.
· 2 min read