Expected Value (Leaving Cert Mathematics): Revision Notes

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

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

---

Worked Examples

---

---

Summary

• Expected Value Formula:
  • For discrete random variables: $E(X) = \sum (x \times P(x))$
  • For continuous random variables: $E(X) = \int x \times f(x) \, dx$
• EV measures the average outcome over many trials.
• Applications include decision-making, fair games, and predictive modelling.
• The expected value may not match any single outcome; it represents the average result over time.