Variance and Standard Deviation

Context: FIT1058_MOC · measures of spread of a Random Variable around its mean · variance $=E((X-\mu {)}^2)$ · bounded deviation via Chebyshev

Quick Revision

• 🎯 Objective: spread of $X$ around $\mu$ ➔ $\mathrm{Var}(X)=E((X-\mu {)}^2)=E(X^2)-E(X{)}^2$.
• 📦 Core Components: variance ➔ $\sigma =\sqrt{\mathrm{Var}}$ (same units) ➔ Chebyshev bound.
• ⚡ Key Constraint: variance adds only for independent variables; Chebyshev is universal but loose.

📝 Core

1. The Measures

• Variance ➔ $\mathrm{Var}(X)=E((X-\mu {)}^2)=E(X^2)-E(X{)}^2$ (average squared deviation).
• Standard deviation ➔ $\sigma =\sqrt{\mathrm{Var}(X)}$ — same units as $X$.

2. Computing & Combining

• Two forms ➔ ${\sum }_k(k-\mu {)}^2\mathrm{Pr}(X=k)$ or $E(X^2)-{\mu }^2$ (often easier).
• Additive if independent ➔ $\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)$ (unlike expectation, needs independence).

3. Chebyshev’s Inequality

• Universal bound ➔ $\mathrm{Pr}(|X-\mu |\ge t\sigma )\le \frac{1}{t^2}$ for any $X$, any $t>0$.
• No shape needed ➔ holds for every distribution.

Key identities:

$\mathrm{Var}(X)=E((X-\mu {)}^2)=E(X^2)-{\mu }^2,\sigma =\sqrt{\mathrm{Var}(X)}$ $\mathrm{Pr}(|X-\mu |\ge t\sigma )\le \frac{1}{t^2}$

⚖️ Core Decision Matrix

Quantity	Units	Note
variance	squared $X$-units	second moment
$\sigma$	$X$-units	interpretable scale
$\mathrm{Var}(X+Y)$	—	additive if independent
Chebyshev	—	general, loose

When It Flips: squaring weights far-out values heavily, so variance is sensitive to extremes. Chebyshev holds for every distribution (the fallback when shape is unknown); well-behaved distributions admit tighter bounds. Binomial variance $np(1-p)$ follows by additivity over independent trials.

📊 Exam Execution Trace

Manual Execution Trace

Fair die ($\mu =3.5$):

Step / State	$k$	$(k-3.5{)}^2$	weighted
0 (Init)	—	—	—
1	1,6	6.25 each	$\frac{12.5}{6}$
2	2,5	2.25 each	$\frac{4.5}{6}$
3	3,4	0.25 each	$\frac{0.5}{6}$

⚠️ Common Mistakes

• 💡 Variance additivity needs independence ➔ $\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)$ only for independent $X,Y$; expectation adds unconditionally.

🧠 Active Recall

Give the two variance formulas, why $\sigma$ is needed, and the independence caveat.

• Hint: Second moment + units.

Answer

• Short answer: $\mathrm{Var}(X)=E((X-\mu {)}^2)=E(X^2)-{\mu }^2$; $\sigma =\sqrt{\mathrm{Var}}$ restores $X$-units.
• Why: Additivity ➔ $\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)$ only if independent.

State Chebyshev's Inequality and its virtue and limitation.

• Hint: Universal but loose.

Answer

• Short answer: $\mathrm{Pr}(|X-\mu |\ge t\sigma )\le \frac{1}{t^2}$ for any distribution.
• Why: Fallback ➔ holds regardless of shape; specific distributions give tighter bounds.