方差fāngchā

Core Concept

The variance of a random variable measures the spread or dispersion of its values around the expected value (mean).

Definition

For a random variable $X$ with expected value $\mu = E(X)$:

$\text{Var}(X) = E\left[(X - \mu)^2\right] = E(X^2) - (E(X))^2$

Notation

• $\text{Var}(X)$ or $V(X)$ - Variance of $X$
• $\sigma^2$ (sigma squared) - Common symbol for variance
• $D(X)$ - Alternative notation (used in Chinese textbooks)

Two Formulas

Formula 1: Definition

$\text{Var}(X) = E\left[(X - \mu)^2\right] = \sum_{i=1}^{n} (x_i - \mu)^2 \cdot p_i$

Formula 2: Computational (more practical)

$\text{Var}(X) = E(X^2) - (E(X))^2$

Memory aid: "Mean of squares minus square of mean"

Properties of Variance

1. Non-negativity

$\text{Var}(X) \geq 0$

Equality holds only when $X$ is constant.

2. Constant Factor

$\text{Var}(aX) = a^2 \cdot \text{Var}(X)$

Note: The factor is squared.

3. Adding a Constant

$\text{Var}(X + b) = \text{Var}(X)$

Adding a constant doesn't change the spread.

4. Combined Linear Transformation

$\text{Var}(aX + b) = a^2 \cdot \text{Var}(X)$

5. Sum of Independent Variables

If $X$ and $Y$ are independent: $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$

Warning: This does NOT hold for dependent variables!

6. Variance of a Constant

$\text{Var}(c) = 0$

Common Distributions

Distribution	Variance
Bernoulli($p$)	$p(1-p)$
Binomial($n, p$)	$np(1-p)$
Uniform($\{1,...,n\}$)	$\dfrac{n^2-1}{12}$

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus.

Example 1: Basic (Difficulty ★★☆☆☆)

A random variable $X$ has the distribution:

$X$	0	1	2
$P$	0.3	0.5	0.2

Find $\text{Var}(X)$.

Solution:

First, find $E(X)$: $E(X) = 0(0.3) + 1(0.5) + 2(0.2) = 0 + 0.5 + 0.4 = 0.9$

Find $E(X^2)$: $E(X^2) = 0^2(0.3) + 1^2(0.5) + 2^2(0.2) = 0 + 0.5 + 0.8 = 1.3$

Apply formula: $\text{Var}(X) = E(X^2) - (E(X))^2 = 1.3 - 0.81 = 0.49$

Answer: $\text{Var}(X) = 0.49$

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Example 2: Intermediate (Difficulty ★★★☆☆)

If $\text{Var}(X) = 4$, find $\text{Var}(3X + 2)$.

Solution:

Using the property $\text{Var}(aX + b) = a^2 \cdot \text{Var}(X)$: $\text{Var}(3X + 2) = 3^2 \cdot \text{Var}(X) = 9 \times 4 = 36$

Answer: $36$

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Example 3: Intermediate (Difficulty ★★★☆☆)

A fair coin is tossed 100 times. Let $X$ be the number of heads. Find $\text{Var}(X)$.

Solution:

$X$ follows Binomial($n=100$, $p=0.5$).

$\text{Var}(X) = np(1-p) = 100 \times 0.5 \times 0.5 = 25$

Answer: $\text{Var}(X) = 25$

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Example 4: Advanced (Difficulty ★★★★☆)

If $E(X) = 2$ and $E(X^2) = 8$, find $\text{Var}(2X - 3)$.

Solution:

First, find $\text{Var}(X)$: $\text{Var}(X) = E(X^2) - (E(X))^2 = 8 - 4 = 4$

Then: $\text{Var}(2X - 3) = 2^2 \cdot \text{Var}(X) = 4 \times 4 = 16$

Answer: $16$

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Example 5: Advanced (Difficulty ★★★★★)

If $X$ and $Y$ are independent with $\text{Var}(X) = 2$ and $\text{Var}(Y) = 3$, find $\text{Var}(2X - 3Y + 1)$.

Solution:

Using properties: $\text{Var}(2X - 3Y + 1) = \text{Var}(2X - 3Y)$ $= \text{Var}(2X) + \text{Var}(-3Y)$ (independence) $= 4\text{Var}(X) + 9\text{Var}(Y)$ $= 4(2) + 9(3) = 8 + 27 = 35$

Answer: $35$

Standard Deviation

The standard deviation is the square root of variance: $\sigma = \sqrt{\text{Var}(X)}$

Standard deviation has the same units as $X$, making it more interpretable.

Variance vs. Expected Value

Property	Expected Value $E(X)$	Variance $\text{Var}(X)$
Measures	Center (location)	Spread (dispersion)
Linear transform	$E(aX+b) = aE(X)+b$	$\text{Var}(aX+b) = a^2\text{Var}(X)$
Sum	Always additive	Additive only if independent

Common Mistakes

❌ Mistake 1: Forgetting to Square the Coefficient

Wrong: $\text{Var}(3X) = 3 \cdot \text{Var}(X)$ ✗

Correct: $\text{Var}(3X) = 9 \cdot \text{Var}(X)$ ✓

❌ Mistake 2: Adding Variances of Dependent Variables

Wrong: If $X$ and $Y$ are dependent, $\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y)$ ✗

Correct: This only works for independent variables ✓

❌ Mistake 3: Confusing Variance with Standard Deviation

Wrong: Standard deviation of Binomial($n,p$) is $np(1-p)$ ✗

Correct: Variance is $np(1-p)$; standard deviation is $\sqrt{np(1-p)}$ ✓

Study Tips

1. ✅ Use computational formula: $E(X^2) - (E(X))^2$ is usually easier
2. ✅ Remember coefficient is squared: $\text{Var}(aX) = a^2\text{Var}(X)$
3. ✅ Check independence: Variance of sum only adds for independent variables
4. ✅ Know binomial variance: $np(1-p)$ is frequently tested

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💡 Exam Tip: When computing variance, always find $E(X)$ and $E(X^2)$ separately first. The computational formula is less error-prone than the definition!