# 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

## Related Questions

### SIR epidemic model with vital dynamics

0  Asked on November 9, 2021

### How to evaluate Euler-type integral

1  Asked on November 9, 2021

### Reference for a bilinear form lemma

0  Asked on November 9, 2021

### Finding a conformal map $T$ such that $T(mathbb{C}-[0,1])subset B(0,1)$

2  Asked on November 9, 2021

### Understanding a detail in a proof related to almost disjoint families (Kunen)

1  Asked on November 9, 2021 by robson

### How to prove this matrix is positive semi-definite?

1  Asked on November 9, 2021 by tree23

### Properties of a certain divisor of the $m^{rm th}$ root of unity in the $m^{rm th}$ cyclotomic ring.

0  Asked on November 9, 2021

### Let $f:Asubset mathbb{R}^nrightarrow mathbb{R}^n$ an submersion of class $C^1$. Show that $f$ is a local diffeomorphism at each point in open A.

0  Asked on November 9, 2021 by user810255

### Is the full subcategory of injectives reflective?

1  Asked on November 6, 2021 by sampah

### Value of $frac{partial }{partial x}left(fleft(x,yright)right)$ at $(0,1)$

2  Asked on November 6, 2021 by ikigai

### Why is MA not provable from ZFC?

1  Asked on November 6, 2021 by grinsekotze

### How to parametrize the intersection of an ellipsoidal surface and a sphere?

0  Asked on November 6, 2021

### To find supremum of this

4  Asked on November 6, 2021 by jessica-griffin

### Can this underdetermined equation be solved?

0  Asked on November 6, 2021 by lonewolf

### Find a sum of fractional series

1  Asked on November 6, 2021 by manabou11

### Unusual ways of summing well-known series — for example, this unusual summation of the geometric series

2  Asked on November 6, 2021

1  Asked on November 6, 2021

### How to compute the series: $sum_{n=0}^{infty} (-1)^{n-1}binom{1/2}{n}$

1  Asked on November 6, 2021 by portokranto

### Removing parameter from set of orthogonal trajectories

1  Asked on November 6, 2021 by rahul-silva

### How to solve quadratic equations with three variable?

0  Asked on November 6, 2021 by user6943953