# Random variable and independence on unit interval

Mathematics Asked by mathim1881 on October 31, 2020

I have a simple probability question that has been confusing me.

Lets say I generate a random variable $$X$$ drawn from the uniform distribution on $$[0,1]$$. Now, lets define two real (nonrandom) numbers $$p,qin[0,1]$$.
Define event $$A$$ to be the event that $$Xleq p$$.
Define event $$B$$ to be the event that $$X geq (1-q)$$.

My question is, for what values of p and q are A and B independent?

My idea is that these two events will be dependent when there is a nonempty intersection between $$Xleq p$$ and $$X geq (1-q)$$ and independent if this intersection is empty. So my answer is that $$A$$ and $$B$$ are independent when $$p-(1-q)leq 0$$. I was hoping someone could help me know if I am correct and if this equality in my inequality answer relating $$p$$ and $$q$$ should be strictly less than? My confusion with this arises because of this boundary when $$p=1-q$$.

By definition, $$A$$ and $$B$$ are independent if and only if $$mathbb P(A cap B) = mathbb P(A) mathbb P(B)$$. In this case, $$mathbb P(A) = p$$, $$mathbb P(B) = q$$, $$mathbb P(A cap B) = p - (1-q) = p+q-1$$ if $$p ge 1-q$$, $$0$$ otherwise. So for independence you need $$p q = p+q-1$$ or $$p q = 0$$. But $$pq - (p+q-1) = (p-1)(q-1)$$. Thus the condition is that $$p=0$$ or $$p=1$$ or $$q=0$$ or $$q=1$$.

Answered by Robert Israel on October 31, 2020

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