Expected Value vs Probability Distribution

The expected value of a random variable $X$ is written $E(X)$ (E of X) or $\mu_X$ (mu sub X), and it is the probability-weighted average of the possible values:

Each value is multiplied by its own probability rather than counted once, and that weighting is the whole difference between an expected value and an ordinary average.

Let $X$ be the number of cars a household owns, with $P(0) = 0.2$, $P(1) = 0.4$, $P(2) = 0.3$ and $P(3) = 0.1$. Then $E(X) = 0(0.2) + 1(0.4) + 2(0.3) + 3(0.1) = 1.3$ cars.

That 1.3 carries two lessons. First, an expected value need not be a value the variable can actually take: no household owns 1.3 cars. It describes the average over many households, not a prediction about the next one. Second, it is not the most likely outcome either. Owning 1 car is the most probable single result here, and most likely and average are separate questions. The mode answers the first, the expected value answers the second.

The other frequent slip is dropping the weights entirely. The plain average of 0, 1, 2 and 3 is 1.5, and that would be correct only if all four outcomes were equally likely. They are not. Weight only moves the average through the value it sits on, so the 0.2 parked on 0 contributes nothing at all: $0(0.2) = 0$. The pull comes from the top of the table, where 3 carries only 0.1 of the weight instead of the 0.25 a flat average assumes, and that is what brings the expected value down to 1.3.

Some distributions come with a shortcut. A binomial variable with $n$ trials and success probability $p$ has $\mu_X = np$ directly, so 10 free throws shot at $p = 0.3$ give an expected 3 makes without summing eleven separate terms. The law of large numbers is what makes any of this useful in practice: as the number of repetitions grows, the observed sample mean converges on $E(X)$. That also fixes the relationship worth keeping straight. $E(X)$ is a parameter of the probability distribution, a fixed number you calculate; a sample mean is a statistic that varies from one batch of data to the next and settles toward it.

The sum here is the discrete version. For a continuous variable the same weighted-balance idea becomes an integral of $x$ against the density instead of a sum, which is the calculus notion of the average value of a function.

The same weighted sum, with dollar payoffs, is how expected value is calculated in economics: how to calculate expected value.

A probability distribution says how the total probability of 1 is divided among the values a random variable can take. For a discrete variable it is a table or a formula giving $P(X = x)$ for every possible $x$, and two conditions define it: each probability satisfies $0 \le P(X = x) \le 1$, and $\sum P(X = x) = 1$. For a continuous variable there is no table. Probability comes from area under a density curve, and the area under the whole curve is 1.

Let $X$ be the number of people in a car passing a checkpoint, with $P(1) = 0.35$, $P(2) = 0.30$, $P(3) = 0.20$, and the last probability left blank. The sum condition fills it in: $P(4) = 1 - (0.35 + 0.30 + 0.20) = 0.15$. That is not a trick, it is the definition doing the work, and it is also why a table whose entries add to 0.97 is not a probability distribution at all.

The misreading to watch is "I rolled the die 60 times and got twelve 3s, so $P(X = 3) = 0.20$." That is a relative frequency from one sample, not the distribution. A probability distribution is what the chance process does over the long run and it is fixed before any data exist; a sample wobbles around it and settles closer as the number of trials grows.

The discrete and continuous cases part company over a single value. For the car table, $P(X \le 2) = 0.65$ while $P(X < 2) = 0.35$, because $X = 2$ carries real probability of its own. For a continuous variable, $P(X = x) = 0$ at every single $x$, so $P(X \le 2)$ and $P(X < 2)$ are equal. Reading a strict inequality as if it were the other kind is a genuine error for counts and harmless for measurements.

Probability distributions enter the course at topic 2.8, Introduction to Random Variables and Probability Distributions.