Q. 2.9 - A First Course in Probability

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$1, 2, and 3$.

Step-by-Step Solution

Answer

Therefore,

       f(·) satisfies Axioms 1, 2, and 3.

Hence Proved.

Step 1 Given information.

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

Step 2 Explanation.

$f\left(E\right)=n\left(E\right) / n$. Axiom$1$. By definition$n(·)$ is a counting function. Therefore, $f(·)$ 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,$\Omega$ 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${\left\{E_m\right\}}_{m=1}^{\infty }$.