Mathematical Expectation

When you participate in a game of chance, buy insurance, or invest in the stock market, you are facing a scenario with multiple possible outcomes, each with its own probability. How do you decide if a risk is mathematically "worth it"?

The answer lies in Mathematical Expectation (or Expected Value). It is a fundamental concept in probability that calculates the long-run average outcome of a random process if it were repeated infinitely many times.

1. What is Expected Value?

The Expected Value, denoted as $E(X)$ or sometimes $\mu$ (mu, representing the mean), is the probability-weighted average of all possible values a random variable can take.

• Key Misconception: The "expected" value does not mean it is the most likely value to occur on a single trial. In fact, for many distributions, the expected value isn't even a possible outcome!
  • Example: The expected value of rolling a standard die is 3.5. You cannot actually roll a 3.5. However, if you roll a die 1,000 times and average the results, that average will be extremely close to 3.5.

Formula for Discrete Random Variables

For a discrete random variable $X$ with possible values $x_1, x_2, \dots, x_n$ and corresponding probabilities $P(x_i)$, the expected value is:

$$ E(X) = \sum [x_i \cdot P(x_i)] $$

• In English: Multiply every possible value by its corresponding probability, and then add all those products together.

Example: A Simple Betting Game

You pay $5 to play a game where you flip a fair coin. If it lands Heads, you win $10. If it lands Tails, you win $0. What is your expected profit?

1. Define the Random Variable $X$ (Profit):
   • Outcome 1 (Heads): You win $10, but paid $5, so profit = +$5. Probability = 0.5.
   • Outcome 2 (Tails): You win $0, but paid $5, so profit = -$5. Probability = 0.5.
2. Calculate $E(X)$: $$ E(X) = (5 \times 0.5) + (-5 \times 0.5) = 2.5 - 2.5 = 0 $$
3. Interpretation: The expected profit is $0. This is called a fair game because neither the player nor the house has a mathematical advantage in the long run.

2. Properties of Expectation

Mathematical expectation behaves linearly, which makes calculations involving multiple variables or scaled variables very straightforward.

Let $a$ and $b$ be constants, and $X$ and $Y$ be random variables.

1. Expectation of a Constant: $E(c) = c$ (If an outcome is always exactly 5, its average is 5).
2. Scaling and Shifting: $E(aX + b) = a \cdot E(X) + b$ (If you multiply every prize in a game by 2, your expected winnings also multiply by 2).
3. Addition of Variables: $E(X + Y) = E(X) + E(Y)$ (The expected value of rolling two dice is simply the expected value of the first die plus the expected value of the second die).

3. Variance and Standard Deviation

While Expected Value tells us the "center" or long-term average, it doesn't tell us the whole story. We also need to know how spread out or risky the outcomes are.

Consider two games: * Game A: 100% chance to win $10. ($E(X) = 10$) * Game B: 50% chance to win $100, 50% chance to lose $80. ($E(X) = 10$)

Both games have the same expected value, but Game B is far more volatile. We measure this volatility using Variance and Standard Deviation.

Variance, $Var(X)$ or $\sigma^2$

Variance measures the average squared distance between the possible outcomes and the expected value (mean).

Formula: $$ Var(X) = E[(X - \mu)^2] $$ A much more common and easier computational formula is: $$ Var(X) = E(X^2) - [E(X)]^2 $$ (The expected value of the squares MINUS the square of the expected value).

Standard Deviation, $SD(X)$ or $\sigma$

Because Variance is squared, its units are also squared (e.g., "squared dollars"), which is hard to interpret. To fix this, we take the square root of the variance to get the Standard Deviation.

Formula: $$ \sigma = \sqrt{Var(X)} $$

Standard Deviation provides a measure of risk or spread in the same units as the original random variable. A higher Standard Deviation means higher risk and wider spread.

4. Properties of Variance

Unlike Expected Value, Variance is not perfectly linear.

Let $a$ and $b$ be constants. 1. Variance of a Constant: $Var(c) = 0$ (A constant never changes, so it has zero spread/volatility). 2. Shifting does nothing: $Var(X + b) = Var(X)$ (Adding $5 to every payout shifts the average, but the spread between the payouts remains exactly the same). 3. Scaling is squared: $Var(aX) = a^2 \cdot Var(X)$ (If you multiply all payouts by 2, the variance is multiplied by $2^2 = 4$).

Summary

• Expected Value ($E(X)$): The long-run probability-weighted average outcome of a random process.
• Calculated by $\sum [x \cdot P(x)]$.
• Variance ($Var(X)$): Measures the spread or risk of the outcomes.
• Calculated by $E(X^2) - [E(X)]^2$.
• Standard Deviation ($\sigma$): The square root of variance, providing a measure of risk in the original units.