Expected Value

The expected value - also known as the mathematical expectation - is a simple yet powerful way to describe what you can expect on average from a random event.

Think of it this way: if you repeated the same random experiment - like a game, a draw, or a gamble - thousands of times, the expected value tells you how much you'd gain or lose per round on average.

It’s one of the cornerstones of probability theory - and essential for understanding risk and reward.

Note. When calculating expected value, it's important to distinguish between two types of random variables: discrete and continuous. A discrete random variable can take only specific values, like the outcome of rolling a die or the number of customers in line. A continuous random variable, on the other hand, can take on any value within a range - like a person's height or room temperature - and is described using a probability density function.

Expected Value: The Discrete Case

Suppose you have a discrete random variable $V$ that can take on $n$ possible values: $v_1, v_2, ..., v_n$, each with a corresponding probability $p_1, p_2, ..., p_n$, such that:

$$ \sum_{j=1}^n p_j = 1 $$

The formula for the expected value is then:

$$ \mathbb{E}[V] = \sum_{j=1}^n v_j \cdot p_j $$

You're essentially calculating a weighted average, where each value is weighted by how likely it is to occur.

Example. A lottery offers three possible prizes: €100, €50, and €0, with probabilities of 0.1, 0.2, and 0.7 respectively. The expected value is: $$
 \mathbb{E}[V] = 100€ \cdot 0.1 + 50€ \cdot 0.2 + 0€ \cdot 0.7 = 10€ + 10€ + 0€ = 20€ $$ So, on average, each ticket is worth €20 - even though none of the actual prizes is exactly €20.

Keep in mind: the expected value is not a prediction. It’s a long-run average. Many times, it doesn’t match any single outcome.

So why does it matter?

Because it helps you make smarter decisions in uncertain situations.

Example. Imagine a coin toss: if it lands heads, you win €100; if it lands tails, you win nothing. Each has a 50% chance: $$  p(\text{heads}) = 0.5 $$ $$ p(\text{tails}) = 0.5 $$ That gives you: $$ \mathbb{E}[V] = 100€ \cdot 0.5 + 0€ \cdot 0.5 = 50€ $$ On average, each toss is worth €50 - even though you never actually win that amount.

That’s what makes expected value so powerful: it offers an objective measure to guide decisions under uncertainty.

You’ll find it everywhere:

• In finance, it helps assess investment returns.
• In statistics, it's used to estimate population means.
• In game theory, it helps pick optimal strategies.
• In economics, it supports decision-making under risk.

In short: if you're dealing with uncertainty, expected value is your compass.

Expected Value: The Continuous Case

For a continuous random variable $X$, we use an integral instead of a sum:

$$ \mathbb{E}[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx $$

Here, $f(x)$ is the probability density function. The concept is the same: multiply each value by its probability - only now, those probabilities are infinitesimal, and you integrate over the entire domain.

Example. Let’s take a variable $X$ that’s uniformly distributed over the interval $[0, 1]$. The probability density function $f(x)$ is constant and equals 1 throughout that interval, since the total area under the curve must equal 1. To find the expected value, we compute: $$\mathbb{E}[X] = \int_0^1 x \cdot f(x) \, dx = \int_0^1 x \cdot 1 \, dx = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1}{2}$$ So even though the variable can take any value between 0 and 1, the average (expected) value is exactly 0.5 - the midpoint of the interval.

Bottom line? The expected value is like a mathematical north star: it won’t predict what happens next, but it helps you make the best possible decision when the outcome isn’t certain.

It’s one of the most effective tools we have for assessing risk, estimating outcomes, and making smart, data-driven choices in the real world.

And there's much more to explore.