Derivations · Start here
Derivations: how the main results are arrived at, and what they mean
Each page here takes one result from the statistics and economics lessons and shows where it comes from, one small step at a time. Every symbol is defined before it is used, a picture appears where the idea turns, and the page ends on what the result means and where the exams ask it. 7 derivations are open now.
Method
How to read a derivation
A derivation is a chain. It starts from something you already accept (a definition, or a fact about adding and multiplying) and takes small steps, each one justified by the one before, until it reaches the result. Nothing is skipped, and nothing is a trick: if a step looks like a leap, read the line under it for the reason.
On these pages the steps open one at a time. Read a step, say it to yourself in words, and only then open the next. Every formula has a “say it in words” line you can open if the symbols get dense, and a lab or a picture wherever the idea is easier to see than to read.
Symbols
Reading the symbols
The strip below covers the notation that appears most. Tap a symbol to see how to say it, what it tells you to do, and a four-number example. The Σ card has a stepper.
SymbolsSymbols used in the derivationsΣ, x̄, E[ ] and friends: how to say them and what they tell you to do.
Tap a symbol.
Library
The derivations
“Why do we multiply deviations and average them, where does the shortcut E[XY] − E[X]E[Y] come from, and why is zero covariance not the same as independence?”Covariance: what it is, where the formula comes from, and what it meansFrom the question 'do two things move together?' to the formula, drawn as signed rectangles on Ashanagar households; the shortcut, the rules for shifting, scaling and adding, independence against zero covariance, and the bound that gives correlation.“Where does the binomial formula come from, why is its variance npq, and how does the Poisson come out of the binomial?”The binomial and Poisson distributions, derivedThe binomial by counting sequences of independent trials, its mean and variance from indicators, and the Poisson as the limit of the binomial, with a sequence counter, a binomial-to-Poisson lab, and the mean and variance worked out.“Where does the exponential come from, why does it forget how long you have waited, and how does the gamma come from adding exponential waits?”The exponential and gamma distributions, derivedThe exponential as the wait for the first event of a Poisson process, its mean, variance and median, memorylessness, then the gamma as the wait for the r-th event and as a sum of exponentials, with a memoryless lab and a gamma lab with simulated sums.“Where does the bell shape come from, why is there a square root of 2 pi in the formula, and why does standardising let one table serve every normal?”The normal distribution, derivedThe bell as the shape the binomial settles into, the constant 1 over root 2 pi from squaring the integral and slicing it into rings, the mean and variance of the standard normal, and standardising, with a bell lab, a ring lab and a z-score lab.“Where do the chi-square, t and F tables come from, why is the sample variance chi-square with n - 1 degrees of freedom, and why is t heavier-tailed than the normal?”The chi-square, t and F distributions, derivedAll three built from the standard normal: the chi-square as a sum of squares (a gamma), the n - 1 degrees of freedom, the variance interval, the t as a normal over a chi-square and the F as a ratio of chi-squares, with simulation labs for each.“What is a moment generating function, why does differentiating it at zero give the moments, and why does it turn sums of independent variables into products?”Moment generating functions, derivedM(t) = E[e^(tX)] derived and explored, the moments read from its derivatives, the functions of the binomial, Poisson, exponential, gamma, chi-square and normal found, sums of independent variables and uniqueness, with a curve-and-tangent lab and a Poisson-sum lab.“How long do you wait for several successes, why is income lognormal in the middle and Pareto at the top, and where does the 80/20 rule come from?”The negative binomial, lognormal and Pareto distributions, derivedThe negative binomial as the wait for the r-th success, the lognormal as the shape of incomes that compound, with mean above median above mode and a Gini that depends on sigma alone, and the Pareto power-law tail with the share of the top and a Gini of 1/(2 alpha - 1), with three labs and the Ashanagar incomes fitted.
Planned
Coming next
- Fisher information and the Cramér-Rao bound. Why no unbiased estimator can have a variance below a fixed floor, and which estimators reach it.
- The delta method. How the variance of an estimate carries through a smooth function of it, such as a ratio or a log.
- The bivariate normal. Two correlated normal variables at once: the joint density, the conditional mean as a regression line, and why regression to the mean happens.
Quick answers
Quick answers
Why learn derivations for an entrance exam?
Most formula questions can be answered by recall, but the ones that separate candidates change a detail: a difference instead of a sum, a sample instead of a population, a shifted variable. Knowing where a formula comes from lets you adjust it instead of guessing, and it makes the formula much easier to remember.
How do I read a summation sign?
The sigma is a loop. The line under it gives the starting value of the counter, the number on top gives the last value, and the expression to its right is what to add each time round. Write out the terms one after another, then add. The Σ card in the symbols strip lets you step through one.
What is the difference between a derivation and a proof?
On these pages they are the same thing in a friendlier form: a chain of small steps, each following from the one before, that ends at the result. A derivation page also says what each symbol means, shows a picture at the step where the idea turns, and ends with what the result means and where the exams ask it.
Read next
All resources →StatisticsCovariance: what it is, where the formula comes from, and what it meansFrom the question 'do two things move together?' to the formula, drawn as signed rectangles on Ashanagar households; the shortcut, the rules for shifting, scaling and adding, independence against zero covariance, and the bound that gives correlation.StatisticsThe binomial and Poisson distributions, derivedThe binomial by counting sequences of independent trials, its mean and variance from indicators, and the Poisson as the limit of the binomial, with a sequence counter, a binomial-to-Poisson lab, and the mean and variance worked out.StatisticsProbability, from zero to exam levelA module of lessons that starts at "what is a sample space" and ends at the DSE's hardest questions: counting, the rules, conditional probability, Bayes' theorem, random variables, the discrete and continuous distributions, and a final lesson of hard problems, each with its own interactives and graded past-paper sets.