CPA Foundation Level
Quantitative Analysis cheat sheet
Every chapter's key points, formulas and definitions on one page, for the last week before the exam. Each heading links to the chapter that explains it.
Functions and matrix algebra
- A function gives each input exactly one output: a linear function graphs as a straight line with slope m and intercept c, a quadratic as a parabola whose turning point is at x = -b / 2a.
- The discriminant b^2 - 4ac tells how many real roots a quadratic has: two if positive, one if zero, none if negative.
- The product AB exists only when A has as many columns as B has rows, and AB is in general not equal to BA.
- For A = [[a, b], [c, d]], det A = ad - bc and A^-1 = (1 / det A) x [[d, -b], [-c, a]]; a zero determinant means there is no inverse.
- Matrices solve simultaneous equations (x = A^-1 b), Markov steady states (s = s x P) and input-output problems (x = (I - A)^-1 d).
Formulas
- The quadratic formula
- x = [-b + or - sqrt(b^2 - 4ac)] / (2a)
- Determinant and inverse of a 2 x 2 matrix
- For A = [[a, b], [c, d]]: det A = ad - bc; A^-1 = (1 / det A) x [[d, -b], [-c, a]]
- Inverse of a 3 x 3 matrix
- A^-1 = (1 / det A) x adj A, where adj A is the transpose of the matrix of cofactors and det A = a11 C11 + a12 C12 + a13 C13
Key terms
- Cubic function
- One whose highest power is x cubed. Its graph has an S shape with up to two turning points and one point of inflexion, and up to three real roots. Total cost curves in economics are cubic: costs rise steeply at first, flatten as efficiencies are gained, then rise steeply again as capacity is strained, which produces the U-shaped average and marginal cost curves.
- Types of matrix
- The types are:
- Addition and subtraction
- Defined only for matrices of the same order: add or subtract corresponding elements. Multiplying a matrix by a scalar (an ordinary number) multiplies every element by it.
- Multiplication
- The product AB exists only when the number of columns in A equals the number of rows in B; the element in row i and column j of the product is the sum of the products of row i of A with column j of B, element by element. The product of an m x n matrix and an n x p matrix is m x p. Matrix multiplication is not commutative: AB and BA are in general different, and one may exist when the other does not.
- Transposition
- The transpose of A, written A with a superscript T, is obtained by writing the rows as columns, so an m x n matrix becomes n x m; the transpose of a product is the product of the transposes in reverse order.
- Solving simultaneous equations
- A system of linear equations is written as A x equals b, where A holds the coefficients, x the unknowns and b the constants; then x equals A inverse times b. This is how a computer solves the normal equations of regression and the constraints of a linear programme.
- Statistical modelling
- The least-squares estimates of a regression with several explanatory variables are given by a matrix formula, the inverse of X transpose X, times X transpose y, where X is the matrix of observations on the explanatory variables and y the vector of observations on the dependent variable; the multiple regression in the chapter on regression is computed this way.
- General applications
- Recording quantities by product and region and multiplying by a price vector to get revenues by region; multiplying a bill-of-materials matrix by a production plan to get raw material requirements; and, in linear programming, holding the coefficients of a linear programme for the simplex method.
The rest of the sheet covers
- Calculus
- Descriptive statistics
- Probability
- Hypothesis testing and estimation
- Correlation analysis
- Regression analysis
- Time series analysis
- Linear programming
- Decision theory
- Worked examples
- Exam technique and revision
- Glossary