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STATS / STUDY EXAMPLEQUICK REFERENCE

Probability & inference10 essentials · conditions, examples & common traps

01Conditional probability

P(A | B) = P(A ∩ B) / P(B)

Use when B is known; require P(B) > 0. The denominator is the group you are already inside.

02Bayes & total probability

P(A | B) = P(B | A)P(A) / P(B)

For a partition Aᵢ, P(B) = Σ P(B | Aᵢ)P(Aᵢ). Count every route into B; a rare cause can stay unlikely.

03Independence vs. disjoint events

P(A ∩ B) = P(A)P(B)

Independence means knowing one does not change the other. Disjoint events cannot both occur; positive-probability disjoint events are not independent.

04Expected value & variance

E[X] = Σ xp(x); Var(X) = E[X²] − E[X]²

For discrete X, weight each value by its probability. E[aX+b] = aE[X]+b; Var(aX+b) = a²Var(X).

05Adding random variables

Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y)

Expectations always add when finite. Variances add without the covariance term only when covariance is zero; independence is sufficient.

06Binomial model

X ∼ Bin(n,p): μ = np; σ² = np(1−p)

Count successes in n independent trials with the same success probability. P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ.

07Normal scores & sample means

z = (x−μ)/σ; SE(x̄) = σ/√n

The standard error formula needs independent observations with common finite variance. A large-sample normal approximation also needs a suitable CLT.

08Confidence interval for a mean

x̄ ± t* · s/√n (df = n−1)

Use the t method for independent normal data; large samples often permit approximation. A 95% method covers the fixed mean in 95% of repeated samples.

09Hypothesis tests

t = (x̄−μ₀)/(s/√n); reject if p ≤ α

Choose the null, alternative and α first. A p-value measures how extreme the data are under H₀; it is not P(H₀ is true).

10Errors & power

Type I = reject true H₀; power = 1−β

Type II means failing to reject a false null. Failing to reject does not prove no effect. Statistical significance does not measure effect size.

WORKED EXAMPLE 1

Q. 15 of 30 Statistics students also take Linear Algebra. Find P(Linear Algebra | Statistics).

Answer. 0.50. Restrict to the 30 Statistics students: 15/30 = 0.50. The full campus is not the denominator.

Variation. If 10 of 50 take both, given Statistics: 10/50 = 0.20. Same rule, different group size.

WORKED EXAMPLE 2

Q. A fair coin is tossed 10 independent times. Find the mean and variance of the number of heads.

Answer. Mean 5; variance 2.5. Here n=10, p=0.5, so np=5 and np(1−p)=2.5.

Variation. For 20 trials with p=0.2: mean 4, variance 3.2. Use 20×0.2×0.8 for the variance.

WATCH OUTDo not swap P(A | B) with P(B | A). SD describes observations; SE describes an estimator. Check independence before adding variances.

CALC / STUDY EXAMPLEQUICK REFERENCE

Derivatives & integrals10 essentials · rules, conditions & worked variations

01Derivative from first principles

f′(x) = limₕ→₀ [f(x+h)−f(x)]/h

The limit must exist from both sides at an interior point. A corner can be continuous but not differentiable; |x| at 0 is an example.

02Power & elementary rules

(xⁿ)′ = nxⁿ⁻¹; (eˣ)′ = eˣ; (ln x)′ = 1/x

Use each rule only where the function is defined and differentiable. For real ln x, x > 0. Constants differentiate to zero.

03Product & quotient

(uv)′ = u′v+uv′; (u/v)′ = (u′v−uv′)/v²

Require v ≠ 0 for the quotient. Do not multiply derivatives together. Keep the numerator order: derivative of top times bottom, minus the reverse.

04Chain & implicit differentiation

[f(g(x))]′ = f′(g(x))g′(x)

Differentiate the outside, keep the inside, then multiply by the inside derivative. When y depends on x, differentiating y² gives 2yy′.

05Tangent & local approximation

L(x) = f(a)+f′(a)(x−a)

Use near a, where the function is differentiable. The derivative is a rate; the tangent approximation is a predicted function value.

06Critical points & extrema

f′(c) = 0 or f′(c) does not exist

c must be in the domain. For a continuous function on [a,b], compare values at interior critical points and both endpoints for absolute extrema.

07Second derivative test

f′(c)=0: f″(c)>0 min; f″(c)<0 max

A zero second derivative is inconclusive. An inflection point requires a change in concavity; f″=0 alone is not enough.

08Antiderivatives & substitution

∫xⁿ dx = xⁿ⁺¹/(n+1)+C, n ≠ −1

For n=−1, use ln|x|+C on an interval avoiding 0. With u=g(x), replace g′(x)dx by du; change bounds in a definite integral.

09Fundamental theorem & signed area

∫ₐᵇ f(x) dx = F(b)−F(a), F′=f

For continuous f on [a,b], an antiderivative evaluates the integral. The integral is signed area; total geometric area uses |f|.

10Integration by parts

∫u dv = uv−∫v du

Choose u so du simplifies, and dv so v is easy to find. On definite integrals, evaluate uv at the bounds and keep those same bounds.

WORKED EXAMPLE 1

Q. For f(x)=x², find the tangent line at x=3.

Answer. y = 6x−9. Since f′(3)=6 and f(3)=9, y−9=6(x−3). The line shares both the value and slope there.

Variation. For f(x)=x³ at x=2: slope 12 and tangent y=12x−16, because f(2)=8.

WORKED EXAMPLE 2

Q. Evaluate ∫ from 0 to 1 of 2x cos(x²) dx.

Answer. sin(1). Set u=x², so du=2x dx and the new bounds are 0 and 1. Then integrate cos u.

Variation. From 0 to √π instead: sin(π)−sin(0)=0. Signed areas can cancel even when the curve is not zero.

WATCH OUTDo not drop the chain-rule factor or +C. Critical point does not mean extremum. Use radians in trigonometric derivative rules.

LINEAR / STUDY EXAMPLEQUICK REFERENCE

Matrices & linear systems10 essentials · tests, dimensions & worked variations

01Dimensions & multiplication

A: m×n, B: n×p ⇒ AB: m×p

Entry (AB)ᵢⱼ = Σ AᵢₖBₖⱼ: a row meets a column. Inner dimensions must match; AB usually differs from BA.

02Solving Ax=b

Consistent ⇔ no row [0 … 0 | c], c ≠ 0

Row-reduce the augmented matrix. A pivot in every variable column gives a unique solution if consistent; free variables give infinitely many.

03Span & linear independence

Ax=0 has only x=0 ⇔ columns independent

Columns span all of Rᵐ exactly when every row has a pivot. A nonzero solution to Ax=0 is a dependence relation among columns.

04Rank & nullity

rank(A) + nullity(A) = n

n is the number of columns. Pivot columns of the original matrix form a column-space basis; the null space is found from the reduced system.

05Determinant & inverse

2×2: det(A)=ad−bc; invertible ⇔ det(A)≠0

For square A, nonzero determinant means full rank and a unique solution for every b. A row swap flips the determinant sign.

06Transpose & inverse order

(AB)ᵀ=BᵀAᵀ; (AB)⁻¹=B⁻¹A⁻¹

Both formulas reverse order. The inverse rule needs invertible square factors. (A+B)⁻¹ is not generally A⁻¹+B⁻¹.

07Eigenvalues & eigenvectors

Av=λv, v≠0; det(A−λI)=0

For square A, solve for λ, then find the nonzero vectors in the null space of A−λI. A zero eigenvalue means A is singular.

08Diagonalization

A=PDP⁻¹; Aᵏ=PDᵏP⁻¹

An n×n matrix is diagonalizable when it has n independent eigenvectors. P holds those vectors; D holds their matching eigenvalues in the same order.

09Orthogonality & projection

projᵤ(v) = [(v·u)/(u·u)]u, u≠0

Perpendicular vectors have dot product zero. If the columns of Q are orthonormal, QᵀQ=I and projection onto their span is QQᵀv.

10Least squares

AᵀAx̂=Aᵀb; residual b−Ax̂ ⟂ col(A)

Minimize squared error when Ax=b is inconsistent. Full column rank makes AᵀA invertible and the minimizer unique; practical solvers often use QR.

WORKED EXAMPLE 1

Q. For A=diag(2,3), find the eigenvalues and an eigenvector for each.

Answer. λ=2 with (1,0)ᵀ; λ=3 with (0,1)ᵀ. Multiplying scales each coordinate axis by its diagonal entry.

Variation. For diag(1,4), eigenvalues are 1 and 4 with the same two axis directions. Both matrices are invertible.

WORKED EXAMPLE 2

Q. Project v=(3,4) onto u=(1,0). Is the residual perpendicular to u?

Answer. Yes. Projection = (3,0), residual = (0,4), and (0,4)·(1,0)=0. Projection keeps the part along the chosen direction.

Variation. For u=(1,1), projection=(3.5,3.5); residual=(−0.5,0.5). Its dot product with u is again zero.

WATCH OUTCheck dimensions first. Do not use reduced-matrix columns as a column-space basis. Zero is never an eigenvector; a zero eigenvalue is allowed.

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See a complete cheat sheet.

Probability & inference10 essentials · conditions, examples & common traps

01Conditional probability

02Bayes & total probability

03Independence vs. disjoint events

04Expected value & variance

05Adding random variables

06Binomial model

07Normal scores & sample means

08Confidence interval for a mean

09Hypothesis tests

10Errors & power

WORKED EXAMPLE 1

WORKED EXAMPLE 2

Derivatives & integrals10 essentials · rules, conditions & worked variations

01Derivative from first principles

02Power & elementary rules

03Product & quotient

04Chain & implicit differentiation

05Tangent & local approximation

06Critical points & extrema

07Second derivative test

08Antiderivatives & substitution

09Fundamental theorem & signed area

10Integration by parts

WORKED EXAMPLE 1

WORKED EXAMPLE 2

Matrices & linear systems10 essentials · tests, dimensions & worked variations

01Dimensions & multiplication

02Solving Ax=b

03Span & linear independence

04Rank & nullity

05Determinant & inverse

06Transpose & inverse order

07Eigenvalues & eigenvectors

08Diagonalization

09Orthogonality & projection

10Least squares

WORKED EXAMPLE 1

WORKED EXAMPLE 2

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