Question

Suppose that an experiment is performed $n$times. For any event $E$of the sample space, let $n\left(E\right)$denote the number of times that event $E$occurs and define$f\left(E\right)=n\left(E\right)/n$. Show that $f(·)$satisfies Axioms$1,2,and3$.

Short answer

Therefore,

f(·) satisfies Axioms 1, 2, and 3.

Hence Proved.

Step-by-step solution

Given information.

let$n\left(E\right)$denote the number of times that event $E$occurs and define$f\left(E\right)=n\left(E\right)/n$.

Explanation.

$f\left(E\right)=n\left(E\right)/n$. Axiom$1$. By definition$n(·)$is a counting function. Therefore, localid="1649236178436" $f(·)$positive. Axiom$2$. $f\left(\Omega \right)=n\left(\Omega \right)/n=1$( Every time we make the experiment some output is obtained, that is,localid="1649236192535" $\Omega$happened) Axiom$3$. Let ${\left\{E_m\right\}}_{m=1}^{\infty }$be a numerable collection of mutually exclusive events ($E$$\cap E_j=\varnothing \forall i\ne j$), Then:

$f\left(\underset{m=1}{\overset{\infty }{\cup }}{E}_m\right)$$=n\left(\underset{m=1}{\overset{\infty }{\cup }}{E}_m\right)/n$

$\stackrel{(*)}{=}\sum _{m=1}^{\infty }n\left(E_m\right)/n=\sum _{m=1}^{\infty }f\left(E_m\right),$

where the equality $(*)$holds by the mutually exclusive condition of${\left\{E_m\right\}}_{m=1}^{\infty }$.