Summary  
This chapter explains how to calculate the expectation (mean) and variance of a discrete random variable by summing each outcome weighted by its probability and its squared deviation, and shows how to implement code to perform these calculations.

General domain of usage  
Statistical analysis

**Video: Visual Explanation of Expectation and Variance**

This video introduces the concepts of **expectation** and **variance** using real-world analogies. You will see how expected value represents the "average" outcome you would anticipate over many repetitions of a random experiment, such as rolling dice or drawing cards. **Variance** is explained as a measure of how much the outcomes tend to spread out from the expected value, helping you understand the consistency or unpredictability of results in practical scenarios.

## 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 $$1/6$$.

1. **Calculate the expectation:**
   $$
   E[X] = 1 \times \frac{1}{6} + 2 \times \frac{1}{6} + 3 \times \frac{1}{6} + 4 \times \frac{1}{6} + 5 \times \frac{1}{6} + 6 \times \frac{1}{6} = 3.5
   $$

2. **Calculate the variance:**
   - First, compute $$(x_i - E[X])^2$$ 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:
      $$
      \text{Var}(X) = \frac{1}{6}(6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25) = \frac{17.5}{6} \approx 2.92
      $$

# 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)

What is the expected value of the random variable $$Y$$ with outcomes 2, 5, and 10, having probabilities 0.2, 0.5, and 0.3, respectively?

Dive into the foundational concepts of probability theory, exploring its principles, applications, and problem-solving techniques using Python. This course blends theoretical understanding with practical coding exercises and visualizations.

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