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Formulas for Expectation and Variance

The expected value (or expectation) of a discrete random variable $X$ is the long-run average value of repetitions of the experiment it represents. The formula is:

E[X] = \sum_{i} x_i \cdot P(X = x_i)

where $x_i$ are the possible outcomes, and $P(X = x_i)$ is the probability of each outcome.

The variance of a random variable $X$ measures how much the values of $X$ are spread out around the expected value. The formula is:

\text{Var}(X) = E[(X - E[X])^2] = \sum_{i} (x_i - E[X])^2 \cdot P(X = x_i)

Step-by-Step Calculation Example

Suppose you have a random variable $X$ representing the outcome of rolling a fair six-sided die, so $X$ can be 1, 2, 3, 4, 5, or 6, each with probability $1/6$ .

Calculate the expectation:
$E[X] = 1 \times \frac{1}{6} + 2 \times \frac{1}{6} + 3 \times \frac{1}{6} + 4 \times \frac{1}{6} + 5 \times \frac{1}{6} + 6 \times \frac{1}{6} = 3.5$
Calculate the variance:
- First, compute $(x_i - E[X])^2$ $(x_{i} - E [X])^{2}$ for each outcome:
  - For $x_1 = 1$ : $(1 - 3.5)^2 = 6.25$ ;
  - For $x_2 = 2$ : $(2 - 3.5)^2 = 2.25$ ;
  - For $x_3 = 3$ : $(3 - 3.5)^2 = 0.25$ ;
  - For $x_4 = 4$ : $(4 - 3.5)^2 = 0.25$ ;
  - For $x_5 = 5$ : $(5 - 3.5)^2 = 2.25$ ;
  - For $x_6 = 6$ : $(6 - 3.5)^2 = 6.25$ .
- Multiply each by $\frac{\raisebox{1pt}{1}}{\raisebox{-1pt}{6}}$ and sum: $\text{Var}(X) = \frac{1}{6}(6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25) = \frac{17.5}{6} \approx 2.92$


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# Compute expectation and variance for a discrete random variable in Python

outcomes = [1, 2, 3, 4, 5, 6]  # Possible outcomes of a fair die
probabilities = [1/6] * 6       # Each outcome has equal probability

# Calculate expectation (mean)
expectation = sum(x * p for x, p in zip(outcomes, probabilities))

# Calculate variance
variance = sum(((x - expectation) ** 2) * p for x, p in zip(outcomes, probabilities))

print("Expectation (mean):", expectation)
print("Variance:", variance)

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Expectation and Variance