Question

Define a collection of events$E_a,0<a<1$having the property that $P\left(E_a\right)=1$for all$a$but$P\left(\underset{a}{\cap }{E}_a\right)=0$

Hint: Let$X$be uniform over$(0,1)$and define each$E_a$in terms of$X$.

Short answer

Therefore, the collection of events$E_a,0<a<1$having the property that$P\left(E_a\right)=1$for all$a$but$P\left(\underset{a}{\cap }{E}_a\right)=0$

Step-by-step solution

Given information:

let $X$be uniform over$(0,1)$.

Explanation:

Let$X~\cup (0,1)$and$E_a=\left\{X\ne a\right\}$.

Explanation:

Thus,${\forall }_a\in (0,1),P\left(E_a\right)=P(X\ne a)=1$but$P\left(\underset{a}{\cap }{E}_a\right)=P\left(\cancel{0}\right)=0$.