Math & StatsLesson 4 of 5

Probability, Statistics & Inference

Taught in English

Questo esame si sostiene in inglese: le lezioni e le domande sono in inglese. L'interfaccia resta in italiano.

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Key formulas

Expected Value

E[X]=Σ xᵢ·pᵢ

Probability-weighted average of all possible outcomes.

Variance

Var(X)=E[X²]−(E[X])²

Also written as σ². Standard deviation σ = √Var(X) — same units as X.

Variance Algebra

Var(aX+b)=a²·Var(X)

Additive constants vanish. Multiplicative constants are squared. So Var(2X+3) = 4·Var(X).

Pearson Correlation

ρ_XY=Cov(X, Y) / (σ_X·σ_Y)

Always between −1 and +1. Zero means linearly uncorrelated (not necessarily independent).

Bayes' Theorem

P(A|B)=P(B|A)·P(A) / P(B)

Updates the prior P(A) with the likelihood P(B|A) to get the posterior P(A|B).

OLS Slope

β̂=Cov(x, y) / Var(x)

The coefficient that minimises the sum of squared residuals.

Part 1 of 1

Expected Value & Variance

Three numbers describe any random variable:

E[X]=Σ xᵢ·pᵢVar(X)=E[X²]−(E[X])²σ=√Var(X) (same units as the data)

Standard deviation σ is what you report — it lives in the same unit as X, unlike variance which is squared.

Worked example

X = number on a fair die. E[X] = (1+2+3+4+5+6)/6 = 3.5. E[X²] = (1+4+9+16+25+36)/6 = 91/6 ≈ 15.17. Var(X) = 15.17 − 3.5² = 15.17 − 12.25 = 2.92. σ ≈ 1.71.

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Linear Algebra & Constrained Optimization Financial Mathematics: Time Value of Money & Valuation

Probability, Statistics & Inference

Taught in English

Questo esame si sostiene in inglese: le lezioni e le domande sono in inglese. L'interfaccia resta in italiano.

100%

Hold ⌘ / Ctrl and scroll to zoom

SubjectLessonDone

Drag to pan · ⌘/Ctrl + scroll or pinch to zoom · tap a node for detail · +/− to expand a branch

Key formulas

Expected Value

E[X]=Σ xᵢ·pᵢ

Probability-weighted average of all possible outcomes.

Variance

Var(X)=E[X²]−(E[X])²

Also written as σ². Standard deviation σ = √Var(X) — same units as X.

Variance Algebra

Var(aX+b)=a²·Var(X)

Additive constants vanish. Multiplicative constants are squared. So Var(2X+3) = 4·Var(X).

Pearson Correlation

ρ_XY=Cov(X, Y) / (σ_X·σ_Y)

Always between −1 and +1. Zero means linearly uncorrelated (not necessarily independent).

Bayes' Theorem

P(A|B)=P(B|A)·P(A) / P(B)

Updates the prior P(A) with the likelihood P(B|A) to get the posterior P(A|B).

OLS Slope

β̂=Cov(x, y) / Var(x)

The coefficient that minimises the sum of squared residuals.

Part 1 of 1

Expected Value & Variance

Three numbers describe any random variable:

E[X]=Σ xᵢ·pᵢVar(X)=E[X²]−(E[X])²σ=√Var(X) (same units as the data)

Standard deviation σ is what you report — it lives in the same unit as X, unlike variance which is squared.

Worked example

X = number on a fair die. E[X] = (1+2+3+4+5+6)/6 = 3.5. E[X²] = (1+4+9+16+25+36)/6 = 91/6 ≈ 15.17. Var(X) = 15.17 − 3.5² = 15.17 − 12.25 = 2.92. σ ≈ 1.71.

Unlock Full Lesson & Timed Practice Drills

Gain full lifetime access to all 5 lessons in Mathematics & Statistics, interactive MindMaps, step-by-step solved questions, and full timed mock exams for UniMi MEF.

Unlock Full UniMi MEF Pass (€29)

Probability, Statistics & Inference

Expected Value & VarianceExpected Value & Variance

Unlock Full Lesson & Timed Practice Drills

Probability, Statistics & Inference

Expected Value & VarianceExpected Value & Variance

Unlock Full Lesson & Timed Practice Drills

Expected Value & Variance

Expected Value & Variance