Probability question, joint probability distribution

Sailboat's route on map is 6km long. Because of the weather, the real traveled distance of the boat is random variable X (km) and speed Y (km/h). Density functions and sample spaces of these two independent random variables below.

$f(x)=\frac{x-6}{2}, \Omega_x=[6,8], $
$g(y)=\frac{y}{8}, \Omega_y=[3,5]$

What's the probability that the boat arrives to the destination in under 2 hours? (hint: draw a picture of the sample space and events, and choose bounds for the integrals from the picture)

Note: I only need the initial integral with bounds and the final answer.

  • Uhm, I assume the distance is uniform between 6 and 8, and the speed is uniform between 3 and 6, but I don't understand what f and g are.

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  • This took me over 30 minutes to answer. Please offer higher bounties for your future posts, specially if you have a short deadline.

  • There was a mistake in computing the double integral. I just fixed it.

  • This doesn't turn out correct either, which is weird.

  • Did you go through my solution? The solution seems fine to me. That's how the problem should be done.

  • Yeah, I went through your solution, checked everything, and I even went ahead and integrated in order dydx. Got the same answer. Must be a glitch in Moodle :). Do you happen to know if I can close the dispute somehow, this was my first time here, hope you didn't get offended haha.

  • I believe the disputes can not be revered. But since we are in agreement, the judge has an easy decision to make.

  • OK, I finally got the right answer. My hunch was right, your h(x,y) was wrong. h(x,y) was in fact f(x)*g(y). This was then integrated first from x/2 to 5, with respect to y, and on the outside from 6 to 8 with respect to x. This gives us 23/32=0.71875. But to make the judges' lives easier and to not provoke you any further, I let the judges take my $10 and give it to you for the time and effort you put in your solution. Have a nice weekend :)

  • The statement of the problem is not clear then. Note that the arrival time =distance/speed. So it makes more sense to have h=f/g.

The answer is accepted.