# Finding probability involving dependent random variables [closed]

Suppose train on line A arrives in time uniformly distributed between 0 and 4mins, train on line B arrives in time uniformly distributed between 0 and 6 mins, and furthermore the time interval between A and B arrival is uniformly distributed between 0 and 4.

a. What is the probability train on line A arrives first

b. What is the probability you wait less than 2 mins for one of the trains to arrive.

Here's how I approached this question,

Let $$A :$$ arrival time for train on line A $$\sim U(0,4)$$, $$B :$$ arrival time for train on line B $$\sim U(0,6)$$

and it is given that $$A-B \sim U(-4,4)$$

a. $$P(A

b. This is where I'm stuck, $$P(\mathrm{min}(A,B) \leq 2) = 1 - P(A>2, B>2)$$

But I'm not sure how I'd go about computing $$P(A>2, B>2)$$ any help or hint is appreciated, thanks.

• I believe there cannot exist any distribution with the properties you describe.
– whuber
Feb 23 at 17:01
• Interesting, can you elaborate more, what makes you believe that ? Feb 23 at 17:28
• Hint: Write $B=(B-A)+A$ and check whether or not the marginal distribution of $B$ can be uniform. Feb 23 at 18:24
• B = (B-A) + A of course always holds, and @whuber is right you cannot have distributions satisfying these conditions, and it just seems to me that problem statement isn't stated very well Feb 24 at 7:45
• In a linked question the easiest route is to say you should have $E[A-B]=E[A]-E[B]$ by linearity of expectation but here $0 \not = 2-3$ Mar 3 at 22:45