Random Variable and its Probability Distribution

Definition of a Random Variable

A random variable is a real-valued function defined on the sample space of a random experiment -- that is, a rule $X$ that assigns a single real number $X(\omega)$ to every outcome $\omega$ of the experiment. For example, if two coins are tossed, the sample space is $\{HH,HT,TH,TT\}$, and $X=$ "number of heads" assigns $X(HH)=2$, $X(HT)=X(TH)=1$, $X(TT)=0$.

This chapter deals only with discrete random variables: those that take a finite (or countable) set of distinct values $x_1,x_2,\dots,x_n$.

Probability Distribution of a Random Variable

The probability distribution of a discrete random variable $X$ is the table (or function) that lists every value $X$ can take, together with the probability of $X$ taking that value:

where $p_i=P(X=x_i)$.

The Two Defining Conditions

A table of values and numbers is a valid probability distribution if and only if it satisfies both:

The first condition simply says probabilities cannot be negative; the second says the random variable must take some value with certainty. Both conditions must always be checked (or, if a distribution contains an unknown constant, used to solve for it) before any further calculation with the distribution is trusted.

Worked Illustration

Three fair coins are tossed simultaneously and $X$ denotes the number of tails obtained. The sample space has $2^3=8$ equally likely outcomes. Exactly $\binom30=1$ outcome gives $0$ tails, $\binom31=3$ give $1$ tail, $\binom32=3$ give $2$ tails, and $\binom33=1$ gives $3$ tails, so the probability distribution is exactly the table shown alongside this section. Its probabilities sum to $\tfrac18+\tfrac38+\tfrac38+\tfrac18=1$, confirming it is valid. …