Discrete random expectation

The expectation of a random variable is the mean of the results we can expect to get if we measure the random variable ‘infinite’ times.

It’s basically the mean value of the list of possible values multiplied by their individual probabilities.

Notation

We can write the expected value of a list of values X as E(X).

Formula

Suppose we have a list of possible discrete values and their associated probabilities. The value at position n is x_n and the associated probability is p_n.

That means we can write the expectation of a list of discrete values and their probabilities (X) as:

E(X) = \sum x_n p_n

A reminder that the n in the formula just means the position of the value. You don’t need to think about it: it’s just the way we write the formula.

Calculating the expectation

Given that formula, it tells us that we can calculate the expectation of a list of values and their probabilities by multiplying each value by its probability (chance of it happening) and then adding all those products together.

This effectively calculates the mean of the values, weighted by their probabilities. We don’t need to divide by the sum of probabilities, because their sum is always 1 (since they represent all possible outcomes).

Example: Calculating the expectation of a fair die roll

Let’s say we have a standard six-sided die, and we want to calculate the expectation of what we’ll get when we roll it.

The possible values are 1, 2, 3, 4, 5, and 6. Each value has an equal probability of \frac{1}{6} - because it’s a fair dice.

Given that, we can then calculate the expectation using our formula:

• E(X)=\sum x_n p_n
• E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6}
• E(X) = \frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6}
• E(X) = \frac{21}{6}
• E(X) = 3.5

So the expectation of rolling a fair die is 3.5. This means that if we were to roll the die an infinite number of times, the average (mean) value we would get would be 3.5.

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