$E(X) = \mu = \sum_{i} x_i \cdot P(X = x_i)$

This is a weighted average where each value $x_i$ is weighted by its probability. If you repeat the random experiment many times, the average of observed values approaches $E(X)$.

Variance and Standard Deviation

Variance measures the average squared deviation from the mean:

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

The second formula, $E(X^2) - [E(X)]^2$, is often easier to compute:

• Compute $E(X^2) = \sum x_i^2 \cdot P(X = x_i)$
• Subtract $[E(X)]^2$

Standard Deviation is the square root of variance:

$\sigma = \sqrt{\text{Var}(X)}$

Standard deviation measures spread in the same units as $X$, making it more interpretable than variance.

Key Interpretations

• Small $\sigma$: Values cluster near the mean; outcomes are predictable.
• Large $\sigma$: Values spread far from the mean; outcomes are highly variable.
• Rule of Thumb: In many distributions, roughly 68% of values fall within $\mu \pm \sigma$.

Worked Example: Discrete Probability Distribution

Consider a discrete random variable $X$ representing the payout (in dollars) from a lottery ticket with the following distribution:

Step 1: Calculate $E(X)$

$E(X) = 0(0.90) + 5(0.07) + 20(0.02) + 100(0.01) = 0 + 0.35 + 0.40 + 1.00 = 1.75$

So the expected payout is $1.75 per ticket.

Step 2: Calculate $E(X^2)$

$E(X^2) = 0^2(0.90) + 5^2(0.07) + 20^2(0.02) + 100^2(0.01) = 0 + 1.75 + 8.00 + 100 = 109.75$

Step 3: Calculate Variance

$\text{Var}(X) = E(X^2) - [E(X)]^2 = 109.75 - (1.75)^2 = 109.75 - 3.0625 = 106.6875$

Step 4: Calculate Standard Deviation

$\sigma = \sqrt{106.6875} \approx 10.33$

Most tickets return $0 or a modest prize, but the rare$100 win creates high variability (SD ≈ $10.33).

Second Example: Fair Die Roll

Roll a fair six-sided die; let $X$ be the outcome (1–6).

• $E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + \cdots + 6 \cdot \frac{1}{6} = \frac{21}{6} = 3.5$
• $E(X^2) = 1 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + \cdots + 36 \cdot \frac{1}{6} = \frac{91}{6} \approx 15.167$
• $\text{Var}(X) = 15.167 - (3.5)^2 = 15.167 - 12.25 = 2.917$
• $\sigma = \sqrt{2.917} \approx 1.71$

Common Pitfalls

⚠️ Variance vs. Standard Deviation Confusion: Variance is in squared units (e.g., dollars²), while standard deviation is in the original units (dollars). Always report standard deviation when describing spread. Also, don't forget to take the square root of $E(X^2) - [E(X)]^2$ to get $\sigma$.

⚠️ Probability Must Sum to 1: Before calculating, verify that $\sum P(X = x_i) = 1$. If not, you have an error in your probability table.

⚠️ Expected Value ≠ Most Likely Value: The expected value is a weighted average and may not be a value the random variable can actually take. For the die, $E(X) = 3.5$ is not an outcome.

Calculator Tip

💡 TI-84 / TI-Nspire: Enter values in L1 and probabilities in L2. Use 1-Var Stats L1, L2 (with frequency list L2) to compute mean and standard deviation directly. The calculator uses the given probabilities as weights.

Compute $E(X)$ from first few terms: $E(X) = 1(0.4) + 2(0.6)(0.4) + 3(0.6)^2(0.4) + 4(0.6)^3(0.4) + \cdots$ $= 0.4 + 0.48 + 0.432 + 0.3456 + \cdots$ $\approx 2.50 (converges to 1/0.4 = 2.5)$

For geometric distribution, variance and standard deviation: $\sigma^2 = \frac{1-p}{p^2} = \frac{0.6}{0.16} = 3.75$ $\sigma = \sqrt{3.75} \approx 1.94$

This shows high relative variability: $\sigma/\mu \approx 1.94/2.5 \approx 0.78$, reflecting the unpredictability of when the first success occurs.