Expected value Study Guide

📖 Core Concepts Expected value (EV) – weighted average of a random variable’s outcomes. Discrete: $E[X]=\sumi xi pi$. Continuous: $E[X]=\int{-\infty}^{\infty} x\,f(x)\,dx$. Existence – the series (or integral) must converge absolutely; otherwise $X$ has no finite expectation. Positive/negative parts – $X^{+}= \max(X,0)$, $X^{-}= -\min(X,0)$. $E[X]=E[X^{+}]-E[X^{-}]$ (Lebesgue definition). $E[X]$ finite ⇔ $E[|X|]<\infty$. Linearity – $E[aX+bY]=aE[X]+bE[Y]$ (when expectations exist). Monotonicity – If $X\le Y$ a.s., then $E[X]\le E[Y]$. Indicator variable – $1A$ satisfies $E[1A]=P(A)$. Variance – $\operatorname{Var}(X)=E[X^{2}]-(E[X])^{2}$. --- 📌 Must Remember $E[1A]=P(A)$. $|E[X]|\le E[|X|]$. Markov: $P(X\ge a)\le \dfrac{E[X]}{a}$ for $X\ge0$. Chebyshev: $P(|X-E[X]|\ge k\sigma)\le \dfrac1{k^{2}}$. Jensen: convex $f$ → $f(E[X])\le E[f(X)]$. Hölder: $E[|XY|]\le (E[|X|^{p}])^{1/p}(E[|Y|^{q}])^{1/q}$, $\frac1p+\frac1q=1$. Minkowski: $(E[|X+Y|^{p}])^{1/p}\le (E[|X|^{p}])^{1/p}+(E[|Y|^{p}])^{1/p}$. Sample mean $\bar X$ is unbiased: $E[\bar X]=E[X]$. Characteristic function: $\varphiX(t)=E[e^{itX}]$, $E[X]=i^{-1}\varphi'X(0)$. --- 🔄 Key Processes Compute EV for a discrete r.v. List outcomes $xi$ and probabilities $pi$. Multiply each $xi pi$, sum them. Compute EV for a continuous r.v. Identify pdf $f(x)$. Integrate $x f(x)$ over the support. Check existence of EV Verify absolute convergence of $\sumi |xi|pi$ (discrete) or $\int |x|f(x)dx$ (continuous). Apply Markov/Chebyshev Choose $a$ (or $k\sigma$) → plug into inequality. Use Jensen Identify convex (or concave) $f$. Compare $f(E[X])$ with $E[f(X)]$. --- 🔍 Key Comparisons Markov vs. Chebyshev Markov: works for any non‑negative $X$, bounds $P(X\ge a)$. Chebyshev: uses variance, bounds deviation from the mean $P(|X-E[X]|\ge k\sigma)$. Discrete vs. Continuous EV Discrete: sum $\sum xi pi$. Continuous: integral $\int x f(x)dx$. Linear vs. Non‑linear Transformations Linearity: $E[aX+b]=aE[X]+b$. Non‑linear: need Jensen or moment‑generating techniques. --- ⚠️ Common Misunderstandings “Expectation always exists.” → False; divergence of the series/integral gives no finite EV. Confusing $E[X^2]$ with $\operatorname{Var}(X)$. → Variance = $E[X^2]-(E[X])^2$, not just $E[X^2]$. Applying Markov to variables that can be negative. → Markov requires $X\ge0$. Assuming Jensen works for any function. → It only holds for convex (or concave) $f$. --- 🧠 Mental Models / Intuition EV = “center of mass” of the probability distribution. Think of each outcome as a weight placed at $xi$; the EV is where the balance point lies. Markov: If the average height is $h$, the fraction of people taller than $2h$ can’t exceed $1/2$. Chebyshev: Most data cluster within a few standard deviations; the farther you go, the fewer points can be there (inverse‑square law). Hölder/Minkowski: Generalizations of Cauchy‑Schwarz; they tell you how “size” (norm) behaves under multiplication/addition. --- 🚩 Exceptions & Edge Cases Non‑absolute convergence → series may conditionally converge but EV is undefined (e.g., Cauchy distribution). Infinite variance – Chebyshev requires finite variance; heavy‑tailed distributions (Cauchy) invalidate it. Indicator expectation – works for any event, even if the event has probability 0 or 1. --- 📍 When to Use Which Use Markov when you have a non‑negative r.v. and only the mean is known. Use Chebyshev when variance is known and you need a bound on deviation from the mean. Use Jensen to bound expectations of convex/concave transformations (e.g., $E[e^{X}]$). Use Hölder for products of random variables when you have $p$‑th and $q$‑th moments. Use Minkowski to bound the $p$‑norm of a sum (useful in $L^p$ spaces). --- 👀 Patterns to Recognize “Mean of a sum = sum of means” → whenever a problem involves $E[X+Y]$, immediately apply linearity. “Bound on tail probability” → look for $P(X\ge a)$ or $P(|X-E[X]|\ge t)$ → think Markov or Chebyshev. “Convex function of a r.v.” → Jensen is the go‑to inequality. “Product of random variables” → check Hölder if moments of each factor are known. --- 🗂️ Exam Traps Choosing the wrong inequality: Using Markov on a variable that can be negative yields an invalid bound. Dropping the absolute value in Markov: $P(X\ge a)$, not $P(|X|\ge a)$. Confusing unbiasedness with consistency: Sample mean is unbiased and converges by LLN, but unbiasedness alone doesn’t guarantee convergence for small $n$. Miscalculating variance: Forget the square on the mean term: $\operatorname{Var}(X)=E[X^2]-(E[X])^2$, not $E[X]^2-E[X^2]$. Assuming Jensen works both ways: For concave $f$, inequality reverses: $f(E[X])\ge E[f(X)]$. ---