Expected Value

Overview

The expected value (EV) is a fundamental concept in probability and statistics that represents the long-term average or mean of a random variable over many trials or occurrences. It gives a measure of the central tendency of the variable.

Formulae

The formula for expected value depends on whether the random variable is discrete or continuous:

For a Discrete Random Variable:

E(X) = \sum (x \times P(x))

Where:

$x$ : Value of the random variable.
$P(x)$ : Probability of $x$

For a Continuous Random Variable:

E(X) = \int x \times f(x) \, dx

Where:

$f(x)$ : Probability density function of $x$ The expected value does not necessarily equal one of the possible outcomes; instead, it represents the average outcome over repeated trials.

Applications

Decision-Making: EV is used in economics, insurance, and gambling to evaluate risks and returns.
Fair Games: A game is considered "fair" if the expected value is zero.
Predictive Models: Expected values are used to predict average outcomes in experiments and real-world scenarios.

Worked Examples

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Example 1: Rolling a Fair Die

Problem: Find the expected value of rolling a fair six-sided die.

Solution:

Step 1: Each outcome $(1, 2, 3, 4, 5, 6)$ has an equal probability:

P(x) = \frac{1}{6}

Step 2: Apply the formula:

E(X) = \sum (x \times P(x)) = (1 \times \frac{1}{6}) + (2 \times \frac{1}{6}) + \dots + (6 \times \frac{1}{6})

Step 3: Simplify:

E(X) = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5

Answer: The expected value is 3.5.

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Example 2: Tossing a Biased Coin

Problem: A biased coin has a 70% chance of landing heads. The outcomes are assigned values: 1 for heads and 0 for tails.

Find the expected value.

Solution:

Step 1: Assign probabilities:

$P(\text{Heads}) = 0.7$
$P(\text{Tails}) = 0.3$