Define a collection of eventsEa,0<a<1 having the property that P(Ea)=1for alla butP(∩aEa)=0Hint: LetX be uniform over(0,1) and define eachEa in terms ofX.

Q 5.6 - A First Course in Probability

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

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

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Answer

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

1Step 1 Given information:

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

2Step 2 Explanation:

Let $X ~ \cup (0, 1)$ and $E_{a} = \{X \neq a\}$ .

3Step 3 Explanation:

Thus, $\forall_{a} \in (0, 1), P (E_{a}) = P (X \neq a) = 1$ but $P (\underset{a}{\cap} E_{a}) = P (0) = 0$ .