Trial (Probability Theory)

probability-theory

Definition

Trial

A trial is one performance of a random experiment. If the experiment $\mathcal{E}$ has sample space $\Omega$, its $i$-th trial produces exactly one realised outcome ${\omega }_i\in \Omega$:

$$\mathcal{E}\stackrel{\text{trial }i}{\to }{\omega }_i,{\omega }_i\in \Omega .$$

The random experiment specifies the repeatable procedure; a trial is one concrete execution of that procedure.

Coin tossing

“Toss a coin and record its upper face” specifies the random experiment. The fifth toss is one trial, and $H$ is the outcome of that trial if the coin lands heads.

Warning

Repeated trials are not automatically independent or identically distributed. These are additional assumptions about how the trials are related.