Random Variable vs Probability Distribution

A random variable is a rule that attaches a number to every outcome of a chance process. The rule itself is not random. The randomness sits in which outcome occurs, and once the outcome is known the variable has one definite value. Notation carries that distinction: a capital letter, usually $X$, names the rule, and a lowercase $x$ names one value the rule can produce. That is why probabilities are written $P(X = x)$, read as the probability that $X$ takes the particular value $x$.

Flip a fair coin twice. The sample space is HH, HT, TH, TT, each outcome with probability 0.25. Let $X$ count the heads. Then HH maps to 2, HT and TH both map to 1, and TT maps to 0. Four outcomes collapse into three values, and the probabilities follow the map: $P(X = 1) = 0.50$, while $P(X = 0)$ and $P(X = 2)$ are each 0.25.

"Blood type is a random variable" is the error worth naming. The output has to be a number, because everything done with a random variable afterwards is arithmetic. Blood type is a categorical outcome, so it does not qualify. Build a number out of it and you are fine: the count of type O donors among 20 arrivals is a random variable, and counts can be averaged. The related slip is writing $P(x = 2)$ in lowercase, which asks about a value instead of about the variable.

Random variables split into two kinds according to the values they can take. Discrete ones have values you can list, and continuous ones take any value in an interval and get their probabilities from area under a density curve. That split decides everything downstream, including whether you add probabilities or integrate areas to answer a question.

Random variables are introduced at topic 2.8 of Unit 2, and their summaries, the mean $\mu_X$ (mu-X) and the standard deviation $\sigma_X$ (sigma-X), at topic 2.9, Parameters of Random Variables.

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.