Random Variables

A random variable assigns a numerical value to each outcome of a random process, creating a probability distribution.

A probability distribution pairs each outcome with a nonnegative weight totalling one. Discrete outcomes have probability bars whose heights sum to one. A weighted balance marker locates expected value, which need not itself be a possible outcome.

A random variable turns outcomes into numbers so probability questions can be analyzed with algebra and statistics.

Definition

A random variable $X$ is a function that assigns a real number to each outcome in a sample space. The capital letter names the variable; lowercase values such as $x$ represent possible outcomes of that variable.

For two coin tosses, $X$ might count heads. Then outcomes TT, HT, TH, HH map to $0,1,1,2$.

Discrete and continuous variables

A discrete random variable has countable possible values, such as number of absences or goals. A continuous random variable can take any value in an interval, such as time, height, or temperature under an idealized model.

Discrete probabilities are assigned to points; continuous probabilities are represented by areas over intervals.

Worked example: number of heads

Worked example

Toss two fair coins and let $X$ be the number of heads. The equally likely outcomes are TT, HT, TH, HH.

$x$	$0$	$1$	$2$
$P(X=x)$	$1/4$	$1/2$	$1/4$

The probabilities sum to $1$. For example,

P(X\ge1)=P(X=1)+P(X=2)=\frac34.

The value $1$ has higher probability because two sample outcomes map to it.

Transforming a random variable

If $Y=aX+b$, then

$$ E[Y]=aE[X]+b. $$

Variance satisfies

\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X).

Adding a constant shifts outcomes without changing spread; scaling multiplies standard deviation by $|a|$.

Common mistakes

Treating the random variable as the random process itself. It is a numerical mapping of outcomes.

Assigning equal probabilities to variable values because sample outcomes are equal. Multiple outcomes may map to one value.

Accepting probabilities that do not total $1$. Check the distribution.

Interpreting expected value as a guaranteed next outcome. It is a long-run average.

Confusing discrete point probability with continuous probability. Continuous models use interval area.

Quick self-check

What numerical quantity does $X$ represent?
Is it discrete or continuous?
Have all possible values and their probabilities been included?
Are probabilities nonnegative and total $1$?
Do “at least” and “at most” translate to correct inequalities?
Are mean and spread interpreted as model summaries rather than guarantees?