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 $p=1-q$.

One Answer

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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